So, as per the multiple choice ("a, b, c or d") solution type question alluded to above, would EITHER "(4 + sqrt(2))/4" OR "1 + 1/2*sqrt (2)" actually be given as one of the possible answer choices on an exam? Because is so, either of these two possible answers might seem to fairly jump out as the correct one when compared to the other three possible answers, unless those other three answers also appeared to be equally complex at first glance. To suffice, this seems like an overwrought algebra question to include as one of a great many in a timed and standardized exam. Just saying...
You made this more complicated than necessary. An equally valid answer is 1 + 1/2*sqrt (2) and you can get there in 3 steps (Sqrt(8) +1)/sqrt (8) = sqrt (8)/sqrt(8) + 1/sqrt(8) Simplifying you get 1 + 1/sqrt(8) = 1 + 1/2sqrt(2)
Agreed, too much talking about how to do it wrong & never gets to answer. Never spend time on the wrong way, Always refer & APPLY the basic properties (Assoc, Com, Dist, Eq & introduce vinculum). Here numerator tell us how many denominators are added. The rest is confusion. KISS - they will figure out the rest.
@@Gideon_Judges6 nope my simplification is correct. In fact they both are. 1/sqrt(8) = 1/sqrt(4*2) = 1/(2*sqrt(2)) = sqrt(2)/4 Multiply 1/2sqrt(2) by sqrt(2)/sqrt(2) and you get sqrt(2)/4
@@christophermeier8329 you kind of can. The problem is the numerator has two parts to it that are divided by sqrt 8. Here is a quote of my comment to this video I hope it helps: Rewrite: (8 ^ 1/2) / (8 ^ 1/2) + (1) / (8 ^ 1/2) . The first set Cancels out Leaving 1 + (1/8 ^ 1/2) We have a root in the denominator so we multiply the top and bottom by it: 1 + (1 * 8^ 1/2) divided by (8 ^ 1/2)² 1 + 2(2^1/2) / 8 The 1 is NOT affected by the denominator AND we can reduce the second part: 1 + (2^1/2) divided by 4 Technically the answer the professor gave is correct, BUT some (teachers like me) would mark that wrong because it wasn't FULLY reduced. When you have a fraction as an exponent the numerator is a power and the denominator is the root. Also Remember the rule (x + y ) divided by (x) is the same as x divided by x + y divided by x. This is valid since both parts of the numerator are affected by the denominator.
@@christophermeier8329 Because there's more than 1 term in the numerator. You can't simplify if there's still another term, it has to be factorised form. For example 3/2 = (2+1)/2 isn't = 1. But 4/2 = (2x2)/2 = 2. Idk if you got it but you can search for videos explaining better.
Seems far simpler and clearer to just start with breaking the expression into the sum of 2 fractions √8/√8 + 1/√8, which is (1 + 1/√8) and then simplify to 1 + √2/4
...Which is (sqrt(8)+1)/sqrt(8), which evaluates to approximately 1.353553. Either form, or even 1+sqrt(2)/4, can be crunched to approximately 1.353553 equally as easily.
If this is the correct answer, I would suggest the question should be: "Can you simplify to a common rational denominator?" I'm not sure how either the answer given or 1 + sqrt(2) / 4 "solves" the problem.
Well, it was (4 + √2)/4, but that is the simplest form. Not all simplifications give you something a LOT simpler. I think one point he was making here is that by convention, a radical should not be in the denominator. I don't really understand the reason for that (I've seen it in another video) but rules is rules, I guess.
Exactly. It's not a solution at all. Rationalising the denominator is not "simplifying" or "solving". Rationalising the denominator is just rationalising the denominator.
@@Astrobrant2(4+√2)/4 is no simpler than the original expression. Rationalising the denominator is not "simplifying" or "solving". Rationalising the denominator is just rationalising the denominator. You say you don't really understand this convention of not having a radical in the denominator. That's probably because there is no good reason for such a convention. It's just needless dogma. Indeed, in this very video he specifically uses a term that DOES have a radical in the denominator. He multiplies the entire thing by √8/√8, thereby demonstrating that radicals in denominators are not just OK, they can in fact be useful.
@@Astrobrant2 : The form with the "simpler" denominator would be easier to use if long division was the only method available for obtaining a decimal approximation.
I really enjoy your videos and your explanations. I kick off trolls who aren't nice, just saying. These videos helped me a lot and thank you. Ignore the critics - these are awesome 👍
@@jamesharmon4994 His first step was done without any explanation whatsoever. How is a student supposed to know to do that. Is a student chastised for any of the many first steps that could substitute just as well? The teacher leaves the impression that their way is "right". I claim the final answer is NOT simpler.
Thank you. I home school my son, so I love to keep up with my Math Skills. I keep telling him that math in our language is known as " Mathafu" which deriveds from the word "Ma" meaning truth. This is commom in Bantu. It all comes from ancient Kemet, when the Atlantians, Tehuti and the rest of them arrived to teach the Kemites the truths and universal laws of the universe.
@@kingalfred3902 "Ma" what you call math is occult. If there is no limit to infinity ♾️ then zero is also an illusion. Tahiti said, "that which has a beginning has an end. And that which has no beginning has no end." We have totally misunderstood "zero" , that's why your calculator cannot divide by zero. Example: 1÷0 Once you understand this then, you are now knee deep in the unexplainable realm, which you call "Black Magic" or "The God Realm" where the laws of science do not apply.
@@CAustin582 Of course you can have radicals in the denominator. It's perfectly fine. He uses a term with a radical in denominator in this very video. He multiplies the expression by √8/√8, thereby demonstrating that radicals in denominators are not just perfectly fine, they can also be useful. Rationalising the denominator is not "solving" or "simplifying". Rationalising the denominator is rationalising the denominator. If he'd posed the question as "Can you rationalise the denominator?" then there would be no problem. (√8+1)/√8 and (4+√2)/4 are just different but equally valid and correct ways of expressing the same value. This obsession he has with not permitting radicals in denominators is not mathematics. It's just needless dogma.
@@gavindeane3670 Of course it's fine to have radicals in the denominator in your steps or as part of the problem. The point is that it's not accepted as part of the solution.
@@CAustin582 It's not accepted as part of the solution BY THIS GUY. Mathematically it's perfectly acceptable and normal. For example, the world is full of mathematicians and scientists and engineers who know that the sine and cosine of 45 degrees is 1/√2. In this video he starts with an expression of the form (a+b)/c and he ends up with an expression of the form (a+b)/c. Regardless of what he says in the video, he has not in any way simplified the expression. It is exactly as simple as it was to start with. And "solve" is completely inappropriate for this.
@@gavindeane3670 Then you could also argue that 2/6 is just as valid of an answer as 1/3. Having a radical in the denominator doesn't break any rules of math; it's just a common convention for simplification. This guy definitely didn't make it up.
Perfect! If I'm going in a shop to buy some tiles for my bathroom, the seller ask me "how many sqt do you need?" Obviously I'll answer 4+root os 2 divided by 4! It's super usefull and effective!
Multiply numerator and denominator by √8 and it becomes (8 + √8)/8 => 8/8 + √8/8 => 1 + 2√2/8 => 1 +√2/4 which I believe may be MORE "simplified" than (4+√2)/4.
Well, if the expression was part of an equation, such as: '((sqr(8) + x)/sqr(8) + 3x^2 -6 = 0' then simplifying the fractional part, rationalizing the denominator, would make finding the solution a lot easier.
When you say, "Can you solve?," that means, "What is the numerical value of this expression?" In this case, it is 1.3536. The question should be, "Can you reduce this expression?" For some reason, the presenter has an obsessive attachment to the idea that no expression should have a radical in the denominator. It's almost as bad as his near obsession with PEMDAS. In the Real World, where we want to see numbers, it makes absolutely no difference if there is a radical in the denominator or not.
It's not even "Can you reduce this expression?". It's just "Can you rationalise the denominator?". That's literally all he's done. The expression starts in the form (a+b)/c and by the end it's still in the form (a+b)/c.
Multiply top and bottom by sqrt(8). Get 8 as the denominator (rationalize 🤓), multiply out the nominator to 8 + sqrt(8), and go from there. Not yet watched the vid, but I feel a bit more can be done. However, for the purpose of the exercise, I guess, the denominator, I’d give at least 80% credit at this stage 😎
There was a lot of discussion about what algebraic rules could be used in solving this problem.The most common questions concerned dividing the numerator sqrt(8) by the denominator sqrt(8). You can solve the problem this way, but must realize that this single fraction is the combined result of two fractions separate fractions-each with the common denominator of the sqrt(8). So, the combined fraction must be expanded back to the original rational terms with the correct radical terms. This will also ensure that the denominator is rationalized (note that the sqrt(8) in the original equation is not rationalized because its solution include the term sqrt(2). Here’s the solution version that divides the sqrt(8) by the sqrt(8). Just be aware of the rules and it’s easy: 1. Expand the fraction using its common denominator of sqrt(8): Sqrt(8) 1 ---- + ---- Sqrt(8) Sqrt(8) 2. Now simplify the two fractions: 1 1 + ---- Sqrt(8) 3. Simplify the Sqrt(8): 1 1 +. ----- 2*sqrt(2) 4. Simplify by multiplying by one, in this case the Sqrt(2) / sqrt(2): 1 1 Sqrt(2) + ----. * ----- 2*sqrt(2). Sqrt(2) 5. Eliminate or combine terms: 1 Sqrt(2) -- + ---------- 2*sqrt(2)*sqrt(2) (2 * 2 = 4) 6. Combine to one fraction using the new common denominator of 4: (4) Sqrt(2) --- + ---- (4) 4 7. Solution: a new common, rationalized denominator of 4. We followed algebraic rules that allowed us to divide the sqrt(8) into itself. 4 + Sqrt(2) -------- 4 Regards, Eric
@@kimobrien.You'll need to try that again. That didn't make any sense. The person you replied to raised a completely valid point. The "answer" in this video is not a solution to or simplification of the original expression. It's just a different way of expressing the same thing.
@@gavindeane3670 You do this type of simplification in preparation for a decimal approximation. Doing the approximation of any root and dividing by a whole number is almost always much simpler than dividing a whole number by the decimal approximation or doing multiply decimal approximations and then multiplication or division. Also the smaller the number under the root sign the easier it is to calculate the approximation. It also provided a standard way to compare numbers like this. When I was in HS we didn't have hand held calculators either.
@@kimobrien. Rationalising the denominator is not simplifying. Rationalising the denominator is rationalising the denominator. It might be a useful thing to do sometimes, but he isn't presenting us with a particular requirement of some operation or calculation we need to do next. You are just assuming that the thing we need to do next is a thing that would be easier without a radical in the denominator. Even if that were true it's still not correct to describe the re-expression of (1+√8)/√8 as (4+√2)/4 or 1+√2/4 as "simplification". It's just expressing the same thing in a different way. In the video he claims we are "solving" or "simplifying" the original expression. That's nonsense. All he needs to do is change the question to "Can you rationalise the denominator?". Then (4+√2)/4 and 1+√2/4 would be correct answers to that question. As the question is currently posed, (4+√2)/4 and 1+√2/4 cannot reasonably be described as "answers" at all.
Gave you a like. What's crazy about Algrebra, you can start off reducing something, but it could be out of order. For example, suppose you decided to reduce both of the sqrt(8) to 2*sqrt(2) first. That is not the right way to solve this but it is a valid algebraic method. In this case it won't be a problem, but in many cases doing something in the wrong order may give you the wrong answer or "stump" you. Yes, memory tricks like PEMDAS, FOIL, SOHCAHTOA are there for simple things, but there are no memory tricks to take any equation and in what order do the "multitude" of things in your "bag of tricks" to reduce or solve it. I aced Algrebra/Geometry/Trigonometry but didn't retain things (swiss cheese) over the years... miss just one of the multitudes of rules and/or miss when to do them and... bingo... error/frustration. Algrebra makes sense to me, but there is just soooo many rules that fade with time (unless you are a mathematician). One would think there could be a single "algorithmic" approach to work any equation, but I started to try that and it was mess. So, sadly, you have get a "feel" on how how to solve something.... and to say I've seen this before - which would require you to do algebra virtually all day long 24/7 for your whole life. And Calculus even adds more rules and ordering. But people score well on the math in SATs becuse all the practice questions used in study guides and SAT prep classes are centered on identifying the "types" of problems and being prepared to solve only those... you've seen that before in study. You can get Ds in Algrebra, but study only the types used on SATs and you can do well. I solved my dilemma by buying a TI-92+ calculator that solves algebra and calculus problems using symbology just like we use in math ("pretty print" as they call it). There has to be a better way to burn EVERY rule and order into the brain... to be flawless in math, you MUST know it all COLD... else you get wrong answers and then what is the point.
This is not an equation so it can't be solved. It is a mathematical expression like 1+2 which can be evaluated. You should have said, 'simplify'. Simplify requires some explanation. Usually includes removing surds from denominators, express in lowest form i.e NOT 2/4. And so on. In mathematics precision is everything. I have even heard clever people refer to Tree(3) as a number rather than an expression that encompasses an algorithm for evaluation. X=tree(tree(3)). No solution. However this could be used in algebraic solutions. It's solution can never be written down in finite time.
Here's an idea. How about getting right to solving (simplifying) the problem instead of showing us 5 ways how NOT to solve it? 95% of those who will take the time to watch the solution akready understand the concepts of identity and distribution, so it's not necessary to show us. For the few who don't , refer them to a separate tutorial about those things, and when they are good with that, THEN they can come back and apply those concepts to this problem. 90% of this video is a waste of time for 95% of the viewers.
No. It equals (8 + √8)/8. You forgot to multiply _both_ of the terms in the numerator by √8. Also, you used the parentheses wrong in your first line. It should be (√8 + 1)/√8
a square root in the denominator is not considered fully simplified in mathematical practice. Instead, you should rationalize the denominator to eliminate the square root. The answer is 1 + sq rt 2 over 4 Sq rt 8 = 2 x sq rt 2
super simple: First step: reduce √8 + 1 to 2√2 + 1. Numerator is now 2√2 + 1 Second step: denominator is irrational due to the root sign so we rationalize it by multiplying the numerator and the denominator by √8 Numerator: (2√2 + 1) * √8 which is 2√16 + 2√2 which reduces further to 8 + 2√2. Now the denominator: √8 * √8 is just 8. Third Step reduce fraction further. 8, 2 , 8 can all be divided by 2 so factor out a 2 and we get the teachers answer of (4 +√2) / 4. We can however reduce this further by "pulling" out 4/4 due to the fact (x + y) / z is the same as x/z + y/z. this gives us the final answer 1 + (√2 / 4). The 1 is NOT affected by the denominator Like I said super simple if you know what to do it should take no more than 30 seconds to do. (This was a step by step explanation so it makes it look like more work than it actually is) Edit Note Corrected to reduce the answer beyond the answer given in the video.
Good explanation for beginners. The truth is math often gets reduced to a series of tricks to squeeze out the answer. If you don't know the tricks it is difficult to intuitively find the answers. Yes, math people call them rules but essentially they are the tricks to solving equations. Students looking for a more straight forward process are often confused when confronted with these situations. Compounding the issue is there may be multiple ways of finding alternate forms of the answers. Often successful math skills come down to "training" the person to answer a problem in a certain way similar to Pavlov teaching his dogs. Not kind but often true.
Irrationals in the denominator are frowned upon, because they are way harder to calculate. Also sqrt(8) can be simplified to 2sqrt(2). So 1+1/sqrt(8)= 1+1/(2sqrt(2)) = 1+sqrt(2)/4
This guy is saying "simplify" when what he really means is "rationalise the denominator". What he's doing here isn't simplification at all. His end point (and your end point) aren't any simpler than the starting point.
Why wouldn't you make it 1 + (√2/4)? Doesn't the distributive property work in subtraction too? (4+√2)/4 would become 4/4 + √2/4, which would reduce to 1 + √2/4. You could look at it as distributing 1/4 times the components of the numerator.
You absolutely can express it as 1+√2/4. Nothing wrong with that at all. It doesn't make sense to describe either 1+√2/4 or (4+√2)/4 as a "solution" to the original expression though. They're all just equivalent ways of saying the same thing.
I got that also. I broke the problem into 2 fractions which resulted in the 1 (like in your answer) with a denominator of 1 also. John's answer just combined both over a single denominator (4).
Only when I have luxury of time then I would waste to watch his lectures and not to the fullest but skipping. However, if he cuts down unnecessarily talking and examples he would be the best of all.
I came here for my daily nap. Wake me when he gets to the problem... Short, and concise is key to NOT confusing your students. I had it figured, then he rambled for ten minutes and now I'm confused(says almost every 14 year old).
What I don't understand is why anyone would bother trying to simplify this in the first place. You're headed for an irrational number no matter what you do. You can write the equation 16 different ways, sure, and that helps you think about the ways to solve it for a whole number I suppose. But the real answer to this is about 1.35... or roughly 4/3. The square root of 8 is roughly 2.82. Photographers know this instinctively.
Good god. I saw the problem and thought “how can this equation be simplified down to a rational number?” 18 minutes I will never get back. I recommend you teach civics or something.
Reduction to the absurd: imagine that the original problem simply was: Simplify SQRT(8). SQRT(80) divided by SQRT(10) would be an unlikely answer. Similarly 2 * SQRT(2) is more likely labelled as simpler, but is it really simpler? If I didn't know the SQRT(2) by memory, then SQRT(2) is just as difficult as SQRT(8). When I know SQRT(2) by memory, is it any worse being in the denominator? What I am trying to say is that mathematics is a science of numerical facts. Rating things on their simplicity is not a science. If we labelled these simplification exercises as a means to restate the problem without the value judgment, without the right vs. wrong judgment, wouldn't that be the spirit of mathematical science?
After 18 minutes of work, the "teacher" comes to a "simplification" that is not any easier than the original. Worse than that, a student who is trying to follow all the steps has no idea that each of the steps is not at all the only step that is required, leaving them thinking that math is a deep and dark mystery. When I first read the problem, I had no idea what the "teacher" would do to make it "simpler". That is, the answer to the original problem has not been found even after 18 minutes. AND, all the math in the problem remains, because if you are not using a calculator, how does the student get the square root of 2? If they are using the calculator, they might as well get the square root of 8. So..."teacher"...if you didn't start off with multiplying by root 8 over root 8, what would you have done? How do you explain the alternate choices to the student?
He's not solving it and he's not simplifying it either. He's just re-expressing the same thing in a different way. Rationalising the denominator is not simplifying. Rationalising the denominator is rationalising the denominator.
How about 1/4 * (4 + sum_(k=0)^∞ ((-1)^k (-1/2)_k (2 )^k )/(k!) ) No square root at all, just an infinite sum. Basic limits! Next challenge is to write it as an integral, no square root allowed.
As you can see by the many answers below, there is no single answer to this question. There are many equivalent ways to re-write the problem. There are many different ways to get to some answer. Is one answer simpler than another? Is one answer simpler than the original problem? Is a calculator still required to actually compute a number. I posed the original problem to my wife, and told her of the first step of multiplying by one. Her first response was like that of any inquisitive student...why would you do that? One person suggests "to rationalize the denominator" which leads to the most obvious question of ....why do you have to do that? If you say "to make it simpler" the obvious question is why would a sqrt 8 in the numerator be better than one in a denominator? , etc.etc. These questions have the pretense that there is a right answer and a right way to get there , making the student feel inadequate when they have attempted something else and when they fail to see that the answer is "simple".
Find the comment section interesting. Very few understand the point of this channel. The point is to explain math in a way so even someone with zero mathing abilities will understand. So many want to show off their minimal knowledge and say nonsense like, “I did this in minus 3 moves. That’s how good I am”. All sorts of pathetic. Just keep on keeping on.
It's doesn't need to be accounted for. The √ symbol means "principal square root of", which is the positive square root. Yes, 4 has two square roots, but √4 refers only to one of them. √4 is 2. If you want to refer to both square roots you put ± in front of the √ symbol.
The video and the comments show the precarious state of mathematics in 2024. The problem does not look for a solution but shall require a simplification, or a rational denominator. The vast majority of comments don't understand the problem either.
![16 divided by [ (9 - 5) squared ] times 16 =? Can You Solve In Your Head?](http://i.ytimg.com/vi/MBlHXVxSuHA/mqdefault.jpg)
![16 divided by [ (9 – 5) squared ] times 16 =? Can You Solve In Your Head?](/img/tr.png)
Congratulations -- you made a simple process very complicated. ----> (4 + sqrt(2))/4
@user-ky5dy5hl4d Occam's razor. Best logic is the most efficient. Efficiency is important in mathematics, physics, and computer programming.
@@Dr_piFrog While true, beginners rarely understand how to be efficient.
So, as per the multiple choice ("a, b, c or d") solution type question alluded to above, would EITHER "(4 + sqrt(2))/4" OR "1 + 1/2*sqrt (2)" actually be given as one of the possible answer choices on an exam? Because is so, either of these two possible answers might seem to fairly jump out as the correct one when compared to the other three possible answers, unless those other three answers also appeared to be equally complex at first glance. To suffice, this seems like an overwrought algebra question to include as one of a great many in a timed and standardized exam. Just saying...
You made this more complicated than necessary. An equally valid answer is 1 + 1/2*sqrt (2) and you can get there in 3 steps
(Sqrt(8) +1)/sqrt (8) = sqrt (8)/sqrt(8) + 1/sqrt(8)
Simplifying you get
1 + 1/sqrt(8) = 1 + 1/2sqrt(2)
Agreed, too much talking about how to do it wrong & never gets to answer. Never spend time on the wrong way, Always refer & APPLY the basic properties (Assoc, Com, Dist, Eq & introduce vinculum). Here numerator tell us how many denominators are added. The rest is confusion. KISS - they will figure out the rest.
I got 1 + sqrt(2)/4. I think perhaps something is wrong in your notation of the final simplified answer.
@@Gideon_Judges6 nope my simplification is correct. In fact they both are.
1/sqrt(8) = 1/sqrt(4*2) = 1/(2*sqrt(2)) = sqrt(2)/4
Multiply 1/2sqrt(2) by sqrt(2)/sqrt(2) and you get sqrt(2)/4
@@markmauldin1327 lol what is sqrt? you mean √ ?
how is 1/2sqrt(2) more simplified than 1/sqrt(8)?? Its actually more complicated
no wonder so many people don't like math....if you need help falling asleep, this is a good vid to watch...
lol come take my math class you will be begging for this guy as a teacher.
Here's why I hate math: 18 min and I still do not know why the sqrt 8s can't be canceled....
@@christophermeier8329 you kind of can. The problem is the numerator has two parts to it that are divided by sqrt 8. Here is a quote of my comment to this video I hope it helps:
Rewrite:
(8 ^ 1/2) / (8 ^ 1/2) + (1) / (8 ^ 1/2) .
The first set Cancels out Leaving 1 + (1/8 ^ 1/2)
We have a root in the denominator so we multiply the top and bottom by it:
1 + (1 * 8^ 1/2) divided by (8 ^ 1/2)²
1 + 2(2^1/2) / 8
The 1 is NOT affected by the denominator AND we can reduce the second part:
1 + (2^1/2) divided by 4
Technically the answer the professor gave is correct, BUT some (teachers like me) would mark that wrong because it wasn't FULLY reduced.
When you have a fraction as an exponent the numerator is a power and the denominator is the root. Also Remember the rule (x + y ) divided by (x) is the same as x divided by x + y divided by x. This is valid since both parts of the numerator are affected by the denominator.
@@christophermeier8329 Because there's more than 1 term in the numerator. You can't simplify if there's still another term, it has to be factorised form.
For example 3/2 = (2+1)/2 isn't = 1. But 4/2 = (2x2)/2 = 2.
Idk if you got it but you can search for videos explaining better.
Too much bla, bla, bla.
exactly. ridiculous wading through this.
Just use the time slider. You're not forced to listen to it.
Solved in 18 seconds, and he talks for 18 minutes.
@Nikioko Believe it or not, there really are people who can't solve this equation. I'm not one of them, but this is not proof they don't exist.
this gentleman is teaching the beginners very nicely , enlighting their mind for ever . ❤
Moha from iran , 👌👋
Seems far simpler and clearer to just start with breaking the expression into the sum of 2 fractions
√8/√8 + 1/√8, which is (1 + 1/√8)
and then simplify to 1 + √2/4
Could you make it any harder.
😂
...Which is (sqrt(8)+1)/sqrt(8), which evaluates to approximately 1.353553. Either form, or even 1+sqrt(2)/4, can be crunched to approximately 1.353553 equally as easily.
How about 1 + (square root of 2 ) divided by 4 ???
I dont get his answer
I get yours
If this is the correct answer, I would suggest the question should be: "Can you simplify to a common rational denominator?" I'm not sure how either the answer given or
1 + sqrt(2) / 4 "solves" the problem.
Well, it was (4 + √2)/4, but that is the simplest form. Not all simplifications give you something a LOT simpler. I think one point he was making here is that by convention, a radical should not be in the denominator. I don't really understand the reason for that (I've seen it in another video) but rules is rules, I guess.
Exactly. It's not a solution at all. Rationalising the denominator is not "simplifying" or "solving". Rationalising the denominator is just rationalising the denominator.
@@Astrobrant2(4+√2)/4 is no simpler than the original expression. Rationalising the denominator is not "simplifying" or "solving". Rationalising the denominator is just rationalising the denominator.
You say you don't really understand this convention of not having a radical in the denominator. That's probably because there is no good reason for such a convention. It's just needless dogma.
Indeed, in this very video he specifically uses a term that DOES have a radical in the denominator. He multiplies the entire thing by √8/√8, thereby demonstrating that radicals in denominators are not just OK, they can in fact be useful.
@@gavindeane3670u put me feelings into words. Thank you.
@@Astrobrant2 : The form with the "simpler" denominator would be easier to use if long division was the only method available for obtaining a decimal approximation.
I really enjoy your videos and your explanations. I kick off trolls who aren't nice, just saying.
These videos helped me a lot and thank you. Ignore the critics - these are awesome 👍
It doesn't require an 18 minute explanation !
It does for those who don't understand it.
Especially for a problem which is solved in less than 18 seconds.
@@Nikioko Your statement isn't true for everyone.
Who says so; speak for yourself.
@@jamesharmon4994 His first step was done without any explanation whatsoever. How is a student supposed to know to do that. Is a student chastised for any of the many first steps that could substitute just as well? The teacher leaves the impression that their way is "right". I claim the final answer is NOT simpler.
I would have separated the numerator into two terms rationalizing the second term and left the answer as 1+ (sqrt(2)/4).
Thank you. I home school my son, so I love to keep up with my Math Skills. I keep telling him that math in our language is known as " Mathafu" which deriveds from the word "Ma" meaning truth. This is commom in Bantu. It all comes from ancient Kemet, when the Atlantians, Tehuti and the rest of them arrived to teach the Kemites the truths and universal laws of the universe.
And your point is ?????
@@kingalfred3902 "Ma" what you call math is occult. If there is no limit to infinity ♾️ then zero is also an illusion. Tahiti said, "that which has a beginning has an end. And that which has no beginning has no end." We have totally misunderstood "zero" , that's why your calculator cannot divide by zero. Example: 1÷0 Once you understand this then, you are now knee deep in the unexplainable realm, which you call "Black Magic" or "The God Realm" where the laws of science do not apply.
4 + sqrt(2)
----------------
4
1 + sqrt(2) ✅
----------
4
(4+ sqrt 2)/4 is not finished.
By partial fraction decomposition it becomes 1 + (1/(2* sqrt 2)).
Like this.
Sqrt 8= 2 * sqrt 2.
((2 * sqrt 2)+1)/(2* sqrt 2)
Decompose, the 2 sqrt 2's cancel,
Leaving (1 + (1/(2*sqrt 2))
Whilst it is true that (√8+1)/√8 is the same as (4+√2)/4, the idea that the first one is a question and the second one is the answer is just silly.
You can't have radicals in the denominator.
@@CAustin582 Of course you can have radicals in the denominator. It's perfectly fine. He uses a term with a radical in denominator in this very video. He multiplies the expression by √8/√8, thereby demonstrating that radicals in denominators are not just perfectly fine, they can also be useful.
Rationalising the denominator is not "solving" or "simplifying". Rationalising the denominator is rationalising the denominator. If he'd posed the question as "Can you rationalise the denominator?" then there would be no problem.
(√8+1)/√8 and (4+√2)/4 are just different but equally valid and correct ways of expressing the same value. This obsession he has with not permitting radicals in denominators is not mathematics. It's just needless dogma.
@@gavindeane3670 Of course it's fine to have radicals in the denominator in your steps or as part of the problem. The point is that it's not accepted as part of the solution.
@@CAustin582 It's not accepted as part of the solution BY THIS GUY. Mathematically it's perfectly acceptable and normal.
For example, the world is full of mathematicians and scientists and engineers who know that the sine and cosine of 45 degrees is 1/√2.
In this video he starts with an expression of the form (a+b)/c and he ends up with an expression of the form (a+b)/c. Regardless of what he says in the video, he has not in any way simplified the expression. It is exactly as simple as it was to start with. And "solve" is completely inappropriate for this.
@@gavindeane3670 Then you could also argue that 2/6 is just as valid of an answer as 1/3. Having a radical in the denominator doesn't break any rules of math; it's just a common convention for simplification. This guy definitely didn't make it up.
Perfect!
If I'm going in a shop to buy some tiles for my bathroom, the seller ask me "how many sqt do you need?" Obviously I'll answer 4+root os 2 divided by 4! It's super usefull and effective!
Multiply numerator and denominator by √8 and it becomes (8 + √8)/8 => 8/8 + √8/8 => 1 + 2√2/8 => 1 +√2/4 which I believe may be MORE "simplified" than (4+√2)/4.
All you get is another formula. What is that in practical terms? Would it not be 1.353....?
Well, if the expression was part of an equation, such as: '((sqr(8) + x)/sqr(8) + 3x^2 -6 = 0' then simplifying the fractional part, rationalizing the denominator, would make finding the solution a lot easier.
This can be simplified in five lines in less than 10 seconds.
When you say, "Can you solve?," that means, "What is the numerical value of this expression?" In this case, it is 1.3536. The question should be, "Can you reduce this expression?" For some reason, the presenter has an obsessive attachment to the idea that no expression should have a radical in the denominator. It's almost as bad as his near obsession with PEMDAS. In the Real World, where we want to see numbers, it makes absolutely no difference if there is a radical in the denominator or not.
🤓☝️
It's not even "Can you reduce this expression?". It's just "Can you rationalise the denominator?". That's literally all he's done.
The expression starts in the form (a+b)/c and by the end it's still in the form (a+b)/c.
Silver hammer. You are perfectly correct. Can you solve what?. You really hammered him. Good work.
John is just wasting our time. He should visit the John with sandpaper
@@harrymatabal8448 Rough crowd in here today...😁😲
Or if you separated the two terms, you could have 1+((1/4)sqrt(2)).
There isn't a solution to an expression. You might simply it if clearly stated rules describing simplest forms are given first.
Multiply top and bottom by sqrt(8). Get 8 as the denominator (rationalize 🤓), multiply out the nominator to 8 + sqrt(8), and go from there. Not yet watched the vid, but I feel a bit more can be done. However, for the purpose of the exercise, I guess, the denominator, I’d give at least 80% credit at this stage 😎
There was a lot of discussion about what algebraic rules could be used in solving this problem.The most common questions concerned dividing the numerator sqrt(8) by the denominator sqrt(8).
You can solve the problem this way, but must realize that this single fraction is the combined result of two fractions separate fractions-each with the common denominator of the sqrt(8).
So, the combined fraction must be expanded back to the original rational terms with the correct radical terms. This will also ensure that the denominator is rationalized (note that the sqrt(8) in the original equation is not rationalized because its solution include the term sqrt(2).
Here’s the solution version that divides the sqrt(8) by the sqrt(8). Just be aware of the rules and it’s easy:
1. Expand the fraction using its common denominator of sqrt(8):
Sqrt(8) 1
---- + ----
Sqrt(8) Sqrt(8)
2. Now simplify the two fractions:
1
1 + ----
Sqrt(8)
3. Simplify the Sqrt(8):
1
1 +. -----
2*sqrt(2)
4. Simplify by multiplying by one, in this case the Sqrt(2) / sqrt(2):
1 1 Sqrt(2)
+ ----. * -----
2*sqrt(2). Sqrt(2)
5. Eliminate or combine terms:
1 Sqrt(2)
-- + ----------
2*sqrt(2)*sqrt(2)
(2 * 2 = 4)
6. Combine to one fraction using the new common denominator of 4:
(4) Sqrt(2)
--- + ----
(4) 4
7. Solution: a new common, rationalized denominator of 4. We followed algebraic rules that allowed us to divide the sqrt(8) into itself.
4 + Sqrt(2)
--------
4
Regards,
Eric
Wouldn't 1 + √(2)/4 be simpler? Seems weird to include the 4/4
I agree
So, what's the final answer? Also, does it really have to be *this complicated*?
Is not a more elegant solution ...... 1+ 8 to the power -1/2 ?
Working with perfect squares, multiplying by sqrt2/sqrt2, would give (sqrt16+sqrt2)/sqrt16 = (4+sqrt2)/4, or 1+(sqrt2)/4?
so, what is the answer ? (mine is 5/4sqrt2 )
I guess I don't understand how the "answer" is really any better than the "problem".
You want to get an integer plus or minus a fraction with whole number in the denominator to get a standard value.
@@kimobrien.You'll need to try that again. That didn't make any sense.
The person you replied to raised a completely valid point. The "answer" in this video is not a solution to or simplification of the original expression. It's just a different way of expressing the same thing.
@@gavindeane3670 You do this type of simplification in preparation for a decimal approximation. Doing the approximation of any root and dividing by a whole number is almost always much simpler than dividing a whole number by the decimal approximation or doing multiply decimal approximations and then multiplication or division. Also the smaller the number under the root sign the easier it is to calculate the approximation. It also provided a standard way to compare numbers like this. When I was in HS we didn't have hand held calculators either.
@@gavindeane3670 The correct answer is 1+√2/4
@@kimobrien. Rationalising the denominator is not simplifying. Rationalising the denominator is rationalising the denominator. It might be a useful thing to do sometimes, but he isn't presenting us with a particular requirement of some operation or calculation we need to do next. You are just assuming that the thing we need to do next is a thing that would be easier without a radical in the denominator. Even if that were true it's still not correct to describe the re-expression of (1+√8)/√8 as (4+√2)/4 or 1+√2/4 as "simplification". It's just expressing the same thing in a different way.
In the video he claims we are "solving" or "simplifying" the original expression. That's nonsense. All he needs to do is change the question to "Can you rationalise the denominator?". Then (4+√2)/4 and 1+√2/4 would be correct answers to that question. As the question is currently posed, (4+√2)/4 and 1+√2/4 cannot reasonably be described as "answers" at all.
Basic surds. Multiply by sqrt8/sqrt8 to rationalise the denominator….all over in less than 20 seconds for a student just starting surds.
I have to ask. What app are you using in this video?
Solve? Is it equation? I don't think so.
P=given formula
P=1+1/rt8
=1+1/2rt2
=1+rt2/4
Solve or simplify? The genius at work again.
Neither. All he's doing is rationalising the denominator. He's not solving or simplifying anything.
@@gavindeane3670 You are another genius.
A really great explanation. I will take your Math Skills Rebuilder and your Pre Calculus courses just for the pleasure of learning.
Ans=√8/√8+1/√8=1+1/√8.
Gave you a like. What's crazy about Algrebra, you can start off reducing something, but it could be out of order. For example, suppose you decided to reduce both of the sqrt(8) to 2*sqrt(2) first. That is not the right way to solve this but it is a valid algebraic method. In this case it won't be a problem, but in many cases doing something in the wrong order may give you the wrong answer or "stump" you. Yes, memory tricks like PEMDAS, FOIL, SOHCAHTOA are there for simple things, but there are no memory tricks to take any equation and in what order do the "multitude" of things in your "bag of tricks" to reduce or solve it. I aced Algrebra/Geometry/Trigonometry but didn't retain things (swiss cheese) over the years... miss just one of the multitudes of rules and/or miss when to do them and... bingo... error/frustration. Algrebra makes sense to me, but there is just soooo many rules that fade with time (unless you are a mathematician). One would think there could be a single "algorithmic" approach to work any equation, but I started to try that and it was mess. So, sadly, you have get a "feel" on how how to solve something.... and to say I've seen this before - which would require you to do algebra virtually all day long 24/7 for your whole life. And Calculus even adds more rules and ordering. But people score well on the math in SATs becuse all the practice questions used in study guides and SAT prep classes are centered on identifying the "types" of problems and being prepared to solve only those... you've seen that before in study. You can get Ds in Algrebra, but study only the types used on SATs and you can do well.
I solved my dilemma by buying a TI-92+ calculator that solves algebra and calculus problems using symbology just like we use in math ("pretty print" as they call it).
There has to be a better way to burn EVERY rule and order into the brain... to be flawless in math, you MUST know it all COLD... else you get wrong answers and then what is the point.
Far too much talking
Do what I do, fast forward using the slider. The talking exists for those who need it.
Simple problem with tooooo much talking. It’s teacher disease when they get old
I agree
@@cindychau5374 then skip past it
Very very very long video!!!!!
Lot of animations, talking, and more animations.
18 minutes!!! Man, it is a problem to solve in 1.5 minutes.
TOO LONG
This is not an equation so it can't be solved. It is a mathematical expression like 1+2 which can be evaluated. You should have said, 'simplify'. Simplify requires some explanation. Usually includes removing surds from denominators, express in lowest form i.e NOT 2/4. And so on.
In mathematics precision is everything.
I have even heard clever people refer to Tree(3) as a number rather than an expression that encompasses an algorithm for evaluation.
X=tree(tree(3)). No solution. However this could be used in algebraic solutions. It's solution can never be written down in finite time.
Here's an idea. How about getting right to solving (simplifying) the problem instead of showing us 5 ways how NOT to solve it? 95% of those who will take the time to watch the solution akready understand the concepts of identity and distribution, so it's not necessary to show us. For the few who don't , refer them to a separate tutorial about those things, and when they are good with that, THEN they can come back and apply those concepts to this problem. 90% of this video is a waste of time for 95% of the viewers.
I don't believe that this site is for people who want to proclaim their mathematical genius to the world, but instead is for dullards like me.
1 + (1/sqrt(8))
=1+ sqrt(8)/8
oops
1 +(sqrt(8)/8)
No. It equals (8 + √8)/8. You forgot to multiply _both_ of the terms in the numerator by √8.
Also, you used the parentheses wrong in your first line. It should be (√8 + 1)/√8
a square root in the denominator is not considered fully simplified in mathematical practice. Instead, you should rationalize the denominator to eliminate the square root.
The answer is
1 + sq rt 2 over 4
Sq rt 8 = 2 x sq rt 2
What I'd like to know is how the answer is the problem simplified. All that work and still the answer is not simplified.
It's not any simpler, is it.
What he's actually done is rationalised the denominator, not simplified.
super simple:
First step: reduce √8 + 1 to 2√2 + 1. Numerator is now 2√2 + 1
Second step: denominator is irrational due to the root sign so we rationalize it by multiplying the numerator and the denominator by √8
Numerator: (2√2 + 1) * √8 which is 2√16 + 2√2 which reduces further to 8 + 2√2.
Now the denominator: √8 * √8 is just 8.
Third Step reduce fraction further. 8, 2 , 8 can all be divided by 2 so factor out a 2 and we get the teachers answer of (4 +√2) / 4.
We can however reduce this further by "pulling" out 4/4 due to the fact (x + y) / z is the same as x/z + y/z. this gives us the final answer 1 + (√2 / 4). The 1 is NOT affected by the denominator
Like I said super simple if you know what to do it should take no more than 30 seconds to do. (This was a step by step explanation so it makes it look like more work than it actually is)
Edit Note Corrected to reduce the answer beyond the answer given in the video.
1 + (sqrt(2)/4)
@@ivankemerovo12 good point thanks.
No one thinks the answer is 1. This was way more confusing than it needed to be.
Good explanation for beginners. The truth is math often gets reduced to a series of tricks to squeeze out the answer. If you don't know the tricks it is difficult to intuitively find the answers. Yes, math people call them rules but essentially they are the tricks to solving equations. Students looking for a more straight forward process are often confused when confronted with these situations. Compounding the issue is there may be multiple ways of finding alternate forms of the answers. Often successful math skills come down to "training" the person to answer a problem in a certain way similar to Pavlov teaching his dogs. Not kind but often true.
So true. I am having problems with Algebra because it seems so arbitrary and impractical.
(√8+1)/√8 equals √8/√8+1/√8 equals 1+1/√8. Why is that not considered fully simplified?
1+(1/(2 sqrt 2))
Irrationals in the denominator are frowned upon, because they are way harder to calculate.
Also sqrt(8) can be simplified to 2sqrt(2).
So 1+1/sqrt(8)= 1+1/(2sqrt(2)) = 1+sqrt(2)/4
This guy is saying "simplify" when what he really means is "rationalise the denominator".
What he's doing here isn't simplification at all. His end point (and your end point) aren't any simpler than the starting point.
You can also change sqrt(8) to 2*sqrt(2):
(2sqrt(2) + 1)(sqrt(2))
-------------------------------
(2sqrt(2) (sqrt(2))
(2sqrt(2)sqrt(2)+(sqrt(2))
-------------------------
(2sqrt(2)(sqrt(2))
(2 * 2) + sqrt(2)
------------------------
(2sqrt(2)sqrt(2))
4 + sqrt(2)
-----------------
4
1 + sqrt(2)
---------- ✅✅✅
4
Why wouldn't you make it
1 + (√2/4)?
Doesn't the distributive property work in subtraction too?
(4+√2)/4 would become 4/4 + √2/4, which would reduce to 1 + √2/4.
You could look at it as distributing 1/4 times the components of the numerator.
You absolutely can express it as 1+√2/4. Nothing wrong with that at all.
It doesn't make sense to describe either 1+√2/4 or (4+√2)/4 as a "solution" to the original expression though. They're all just equivalent ways of saying the same thing.
I got that also. I broke the problem into 2 fractions which resulted in the 1 (like in your answer) with a denominator of 1 also. John's answer just combined both over a single denominator (4).
Only when I have luxury of time then I would waste to watch his lectures and not to the fullest but skipping. However, if he cuts down unnecessarily talking and examples he would be the best of all.
That's why I don't have the patience to watch his channel,it boring,repetitive and wastes my time,his teaching style is unattractive
I really don't know what you are saying. You give us an expression, not an equation and you ask us to solve it? What are you up to?
Simplify, not solve (in the title) please.
"Rationalise the denominator" not "simplify" (and yes, definitely not "solve").
I came here for my daily nap. Wake me when he gets to the problem...
Short, and concise is key to NOT confusing your students. I had it figured, then he rambled for ten minutes and now I'm confused(says almost every 14 year old).
By this video, you clearly explained why nobody should take your lessons. I am fortunate to have had much better teachers than you are.
What I don't understand is why anyone would bother trying to simplify this in the first place. You're headed for an irrational number no matter what you do. You can write the equation 16 different ways, sure, and that helps you think about the ways to solve it for a whole number I suppose. But the real answer to this is about 1.35... or roughly 4/3.
The square root of 8 is roughly 2.82. Photographers know this instinctively.
break it down even further 1 + 1/4 x sqrt 2
The square root of 16 is 4. The square root of 8 is @2.89. One-half of 8 is 4.
Solving for what? Where's the unknown?
Thanks. Good video. I'm still smart at math even though I'm stupid at everything else in life.
I'll bet you're not!
@@nickfraser4599 muted
Isnt it ONE?
There is nothing to “solve “because there is no =. in math, language is important.
The distributive property is taught in third grade. You made a simple problem very complicated.
Verdade
Good god. I saw the problem and thought “how can this equation be simplified down to a rational number?” 18 minutes I will never get back. I recommend you teach civics or something.
√8 = √2×2×2 = 2√2 = √8 + 1/ √8 = 2√2 + 1/2√2 = √2 ko up down multiply 2√2 +1/2√2×√2/√2 = 4 + √2/4 answer
2 sqre root 8 +1
Reduction to the absurd: imagine that the original problem simply was: Simplify SQRT(8). SQRT(80) divided by SQRT(10) would be an unlikely answer. Similarly 2 * SQRT(2) is more likely labelled as simpler, but is it really simpler? If I didn't know the SQRT(2) by memory, then SQRT(2) is just as difficult as SQRT(8). When I know SQRT(2) by memory, is it any worse being in the denominator? What I am trying to say is that mathematics is a science of numerical facts. Rating things on their simplicity is not a science. If we labelled these simplification exercises as a means to restate the problem without the value judgment, without the right vs. wrong judgment, wouldn't that be the spirit of mathematical science?
Can’t have the Square Root in the denominator and the answer is definitely not 1 May I suggest multiplying the top and the bottom by the SR of 8
Square roots in denominators are fine. They can even be useful - as you (and the video) demonstrate when you multiply this by √8/√8
Seeing problems and methods like this makes me wonder how I ever made it through algebra. There must be some other way.
Thank you. I was able to get the right answer.
YEA SURE ...!!!!.....
Yeah, me too, I just read down the comments to find the answer.
Multiply top and bottom by SR8 = (1 +SR8)/8. I failed to simplify it further by realizing that SR8 = 2*SR2
After 18 minutes of work, the "teacher" comes to a "simplification" that is not any easier than the original. Worse than that, a student who is trying to follow all the steps has no idea that each of the steps is not at all the only step that is required, leaving them thinking that math is a deep and dark mystery. When I first read the problem, I had no idea what the "teacher" would do to make it "simpler". That is, the answer to the original problem has not been found even after 18 minutes. AND, all the math in the problem remains, because if you are not using a calculator, how does the student get the square root of 2? If they are using the calculator, they might as well get the square root of 8. So..."teacher"...if you didn't start off with multiplying by root 8 over root 8, what would you have done? How do you explain the alternate choices to the student?
Quanta enrolação para resolver um problema elementar. Fala sem parar
(√8 + 1) / √8
= (8 + √8) / 8
= 1 + 2√2 / 8
= 1 + √2 / 4.
is this a question?
Sir!
You made it ,more complicated.
What about 1 + (1/sqrt(8))?
Simplify sqrt 8 to 2 * sqrt 2
1+ 1/(2 sqrt 2)
That's also fine. You could express the √8 bit as 2√2 if you wanted to as well.
You can't solve this problem but you can simplify by rationalizing the denominator.
He's not solving it and he's not simplifying it either. He's just re-expressing the same thing in a different way.
Rationalising the denominator is not simplifying. Rationalising the denominator is rationalising the denominator.
How about 1/4 * (4 + sum_(k=0)^∞ ((-1)^k (-1/2)_k (2 )^k )/(k!) ) No square root at all, just an infinite sum. Basic limits! Next challenge is to write it as an integral, no square root allowed.
I feel for your class, How can a maths teacher waffle this much? Unbelievable!
Mathematicians are always “rationalizing” their beliefs. Lol😊
I'm so lost. I'm glad it's not a lesson on maps. I'll end up in Antarctica
Thank you
Really? 4 times in a row you needed to reach us that you can't use factors when there is a sum or difference involved.
As you can see by the many answers below, there is no single answer to this question. There are many equivalent ways to re-write the problem. There are many different ways to get to some answer. Is one answer simpler than another? Is one answer simpler than the original problem? Is a calculator still required to actually compute a number. I posed the original problem to my wife, and told her of the first step of multiplying by one. Her first response was like that of any inquisitive student...why would you do that? One person suggests "to rationalize the denominator" which leads to the most obvious question of ....why do you have to do that? If you say "to make it simpler" the obvious question is why would a sqrt 8 in the numerator be better than one in a denominator? , etc.etc. These questions have the pretense that there is a right answer and a right way to get there , making the student feel inadequate when they have attempted something else and when they fail to see that the answer is "simple".
Basically 1 round up.
Oh man I got 1+sqrt2/4 I guess I better keep studying.
Your answer is better!!!
/8+1- /8=
4 + /2 =4
How is sq root of 8, 64? 8 squared is 64.
Find the comment section interesting. Very few understand the point of this channel. The point is to explain math in a way so even someone with zero mathing abilities will understand.
So many want to show off their minimal knowledge and say nonsense like, “I did this in minus 3 moves. That’s how good I am”. All sorts of pathetic. Just keep on keeping on.
One simple question, Given the number 4:
The square roots of 4 are +2 and -2. That needs to accounted for!
It's doesn't need to be accounted for. The √ symbol means "principal square root of", which is the positive square root.
Yes, 4 has two square roots, but √4 refers only to one of them. √4 is 2.
If you want to refer to both square roots you put ± in front of the √ symbol.
So what is the "useful" purpose of this exercise?
1.28868(approx.)
1+V2/4.
The video and the comments show the precarious state of mathematics in 2024. The problem does not look for a solution but shall require a simplification, or a rational denominator. The vast majority of comments don't understand the problem either.
All he needed to do was change the question from "Can you solve?" to "Can you rationalise the denominator?"
1+square root of 2 over 4
It seems very long😂
Only NERDS need this information!!😆
1 + sqrt2 ÷4