Super simple. First we simplify: (4 + √12)² (4 + 2√3)² Next we use foil and combine like terms to get our solution (4 + 2√3)(4+2√3) (4 * 4) + (4 * 2√3) + (4 * 2√3) + (2√3 * 2√3) 16 + 8√3 + 8√3 + (4√9) and 4√9 is the same as 4 * 3 or just 12 16 + 16√3 + 12 28 + 16√3 28 + 16√3 is the final answer since we are NOT allowed calculators and 16√3 is fully simplified.
Simplifying is always very important but in this case there is no benefit to simplify first. Just stay with V12 and simplify at the last step. (V12)² = 12 is easier than (2V3)² = 4 . 3 = 12
Did it in my head in about 20 seconds...1.732050807 is close enough to 1¾ to use to get 28 + a s'kosh' less than 28... So I had slightly less than 56...around 55.72-ish.
@@argonwheatbelly637 we do not put this in decimal form as we cannot be exact with it. It's amazing how many people don't realize that being off a billionth of a percent means the difference between a shuttles crew coming home and then blowing up in that beautiful sky
Alright, let me show you two ways to do this. Let’s start with using the FOIL method: (4 + sqrt(12))^2 Raising a term to the second power is the same as multiplying it by itself. Therefore, = (4 + sqrt(12)) * (4 + sqrt(12)) To multiply two factors together that have two terms each, such as this situation, we can utilize the FOIL Method (First, Outer, Inner, Last), multiplying each term in one factor by the two in the other factor and summing them. In general, (a + b) * (c + d) = a*c + a*d + b*c + b*d Doing this to the above problem, we get: (4 + sqrt(12)) * (4 + sqrt(12)) = 4*4 + 4*sqrt(12) + sqrt(12)*4 + sqrt(12)*sqrt(12) Remember that: sqrt(a) * sqrt(b) = sqrt(a * b) as long as either a or b >= 0 So, doing the math: = 16 + 4*sqrt(12) + 4*sqrt(12) + sqrt(12 * 12) = 16 + 8*sqrt(12) + sqrt(144) (Note that a*sqrt(x) + b*sqrt(x) = (a + b)*sqrt(x) ) = 16 + 8*sqrt(12) + 12 = 28 + 8*sqrt(12) However, note that 12 = 4 * 3. From the rule about multiplying square roots above, we can simplify this further: = 28 + 8*sqrt(4 * 3) = 28 + 8*sqrt(4)*sqrt(3) = 28 + 8*2*sqrt(3) (4 + sqrt(12))^2 = 28 + 16*sqrt(3) The second method is a bit quicker, and utilizes the fact that (a + b)^2 = a^2 + 2ab + b^2, proven below: (a + b)^2 = (a + b)(a + b) = a*a + a*b + b*a + b*b = a^2 + ab + ab + b^2 = a^2 + 2(ab) + b^2 = a^2 + 2ab + b^2 Seeing that we can set a = 4 and b = sqrt(12), we can start at: (4 + sqrt(12))^2 = (4)^2 + 2(4)(sqrt(12)) + (sqrt(12))^2 = 16 + 8*sqrt(12) + 12 And the math will be the same as shown above from here. Keep this fact in mind, as this sort of problem is gonna come up quite often.
Is not the usage of the FOIL method in violation of the rule of PEMDAS because you are executing the exponent before calculating what is inside the parenthesis?
This is the problem with PEMDAS (well, one of the problems). It gives you a really strict, rigid calculation sequence which gives people the impression that they're doing something wrong if they do the calculation in a different sequence. As long as you adhere to the precedence rules in PEMDAS, you can do the calculations in whatever sequence you like.
It's not in violation of PEMDAS because FOIL is a rewriting of part of an equation with an equivalent. The process of rewriting an equation with an equivalent is outside of PEMDAS, and can be done at any time. It's like replacing 2^3 with 2*2*2.
@@gavindeane3670 I understand what you're saying, but the way you said it is confusing. ... I'm not actually being critical, because I can't think of a better way to say it either. I suspect some time after a person really understands the concepts behind PEMDAS and is no longer dependent on the acronym, what you are referring to is probably pretty straightforward. But until then, I suspect it's a difficult concept for people to grasp.
@ Yes, Shay! I looked at the way I had done the problem. I am sorry, I should have typed 8 and √12 close together. It is not √12 + 12. It should be 8√12 + 12. ------ Your method was great too, when you wrote √12 as 2√3. There are different ways to do the problem and it is wonderful that we end up with the same correct answer! -------- I should have typed as follows: (4 + √12)(4 +√12) 16 + 8√12 + 12 16 + 12 + 8√12 28 + 8√12 28 + 8√4 x √3 28 + 8 x 2 x √3 28 + 16√3
2 sqrt of 3 is how many sqrts of 3 you have. Now if you multiply 2 sqrts of 3 by 8 how many sqrts of 3 will you have? 16 sqrts of 3. IOW, regard sqrt of 3 as a unit. 2 sqrt of 3 then is two of those units. Now, multiply those two units by 8. How many units do you have? 16 units. Now, turn those units back into sqrts of 3 and how many of them do you have? 16 sqrts of 3. I hope I made some kind of sense for you.
Perhaps you could structure your examples such that you can assume that your online students have worked through all the previous videos, so you wouldn't have to go back and re-explain basic math concepts in order to solve whatever given problem? Your voice is very soothing to those who are anxious about math, but you seem to have decided at the beginning of each video that you have to assume that your audience knows nothing about math, so you explain every math concept a student needs to understand in order to solve a problem. Perhaps you could number your videos, then refer the students back to the correct numbered video to learn *only* the concepts need to solve this problem? If you were to do this, your explanations would be more concise and to the point, which would not only help the people who are trying to understand, but also you, in that you don't have to start every video with a ten minute explanation of all the math concepts the student would have to understand to solve a given problem. This is just a suggestion. Feel free to reject it out of hand. As an engineer who loves math, I really do enjoy your videos.
An equation is defined as two expressions separated by an equals sign. An expression can have numbers, variables and mathematical operations, but it can also be a simple as a single number (a constant expression). The "?" in the given example is a symbol representing an unknown value ... in other words a variable. Based on those definitions, the given example is an equation which can be solved.
@paulfrank8738 wrong, an equation must have at least one unknown to be determined. The above is a numerical expression to be evaluated. Example, 7 + 9 is not an equation, but 7x + 9 = 12 is. Your math knowledge is lacking, you're not that smart. The ? Does not represent an unknown. 7 + 9 = 16 is not an equation because 7 + 9 does not equal 16 it's IDENTICALLY EQUAL to 16. In other words 7+9 is 16 written in a different form. The video asks to work out the value of a numerical expression.
Your videos are great. But it takes way too long. Get to the point.
Super simple. First we simplify:
(4 + √12)²
(4 + 2√3)²
Next we use foil and combine like terms to get our solution
(4 + 2√3)(4+2√3)
(4 * 4) + (4 * 2√3) + (4 * 2√3) + (2√3 * 2√3)
16 + 8√3 + 8√3 + (4√9) and 4√9 is the same as 4 * 3 or just 12
16 + 16√3 + 12
28 + 16√3
28 + 16√3 is the final answer since we are NOT allowed calculators and 16√3 is fully simplified.
Simplifying is always very important but in this case there is no benefit to simplify first. Just stay with V12 and simplify at the last step. (V12)² = 12 is easier than (2V3)² = 4 . 3 = 12
Did it in my head in about 20 seconds...1.732050807 is close enough to 1¾ to use to get 28 + a s'kosh' less than 28... So I had slightly less than 56...around 55.72-ish.
@@argonwheatbelly637 we do not put this in decimal form as we cannot be exact with it. It's amazing how many people don't realize that being off a billionth of a percent means the difference between a shuttles crew coming home and then blowing up in that beautiful sky
@@panlomito I was doing that step-by-step for those people who don't like to watch his videos because he takes to bloody long to show how it's done
(4+sqrt(12))^2
16 + 12 + 8sqrt(12)
28 + 8sqrt(12)
28 + 16sqrt(3)
= 55.713
(4+2sqrt(3))^2
4(2+sqrt(3))^2
4{4+3+4sqrt(3)}
4{7+4sqrt(3)}
28 + 16sqrt(3)
I’m agree with @tomtke7351, my answer is 55.71
Thanks 🙏 you made me love mathematics so much
Alright, let me show you two ways to do this. Let’s start with using the FOIL method:
(4 + sqrt(12))^2
Raising a term to the second power is the same as multiplying it by itself. Therefore,
= (4 + sqrt(12)) * (4 + sqrt(12))
To multiply two factors together that have two terms each, such as this situation, we can utilize the FOIL Method (First, Outer, Inner, Last), multiplying each term in one factor by the two in the other factor and summing them. In general,
(a + b) * (c + d) = a*c + a*d + b*c + b*d
Doing this to the above problem, we get:
(4 + sqrt(12)) * (4 + sqrt(12)) = 4*4 + 4*sqrt(12) + sqrt(12)*4 + sqrt(12)*sqrt(12)
Remember that:
sqrt(a) * sqrt(b) = sqrt(a * b) as long as either a or b >= 0
So, doing the math:
= 16 + 4*sqrt(12) + 4*sqrt(12) + sqrt(12 * 12)
= 16 + 8*sqrt(12) + sqrt(144) (Note that a*sqrt(x) + b*sqrt(x) = (a + b)*sqrt(x) )
= 16 + 8*sqrt(12) + 12
= 28 + 8*sqrt(12)
However, note that 12 = 4 * 3. From the rule about multiplying square roots above, we can simplify this further:
= 28 + 8*sqrt(4 * 3)
= 28 + 8*sqrt(4)*sqrt(3)
= 28 + 8*2*sqrt(3)
(4 + sqrt(12))^2 = 28 + 16*sqrt(3)
The second method is a bit quicker, and utilizes the fact that (a + b)^2 = a^2 + 2ab + b^2, proven below:
(a + b)^2 = (a + b)(a + b)
= a*a + a*b + b*a + b*b
= a^2 + ab + ab + b^2
= a^2 + 2(ab) + b^2
= a^2 + 2ab + b^2
Seeing that we can set a = 4 and b = sqrt(12), we can start at:
(4 + sqrt(12))^2 = (4)^2 + 2(4)(sqrt(12)) + (sqrt(12))^2
= 16 + 8*sqrt(12) + 12
And the math will be the same as shown above from here. Keep this fact in mind, as this sort of problem is gonna come up quite often.
4(7+4square root 3)
this is the right answer
We can still factor out 4 from the last equation.
4² + (2 . 4V12) + (V12)² = 16 + 8V12 + 12 = 28 + 16V3
28 + 16sqrt(3) ? I tried solving in my head
got it sum of 2 sqs 16 + 4 sr 12 + 4 sr 12 + 12 sr 12 is 2 sr 3 thanks for the fun.
Is not the usage of the FOIL method in violation of the rule of PEMDAS because you are executing the exponent before calculating what is inside the parenthesis?
This is the problem with PEMDAS (well, one of the problems). It gives you a really strict, rigid calculation sequence which gives people the impression that they're doing something wrong if they do the calculation in a different sequence.
As long as you adhere to the precedence rules in PEMDAS, you can do the calculations in whatever sequence you like.
It's not in violation of PEMDAS because FOIL is a rewriting of part of an equation with an equivalent. The process of rewriting an equation with an equivalent is outside of PEMDAS, and can be done at any time. It's like replacing 2^3 with 2*2*2.
@@gavindeane3670 I understand what you're saying, but the way you said it is confusing. ... I'm not actually being critical, because I can't think of a better way to say it either. I suspect some time after a person really understands the concepts behind PEMDAS and is no longer dependent on the acronym, what you are referring to is probably pretty straightforward. But until then, I suspect it's a difficult concept for people to grasp.
@@paulfrank8738 Thank you for that.
Thanks
I got 56.25 doing it in my head. Had to approximate square root of 12. Took maybe 30 seconds. I'm 74 and not a mathematician.
28+16square root3
16sqrt3+28
28+ 16 square root of 3
Wasn’t quick but I did get there.
(4+ √12)^2
(4+ √12)(4+ √12)
16+8 √12+12
28+8 √4* √3
28+8*2* √3
28+16 √3
wrong √12 + 12 if this were an actual exam that would be no points.
@
Yes, Shay! I looked at the way I had done the problem.
I am sorry, I should have typed 8 and √12 close together.
It is not √12 + 12.
It should be 8√12 + 12.
------
Your method was great too, when you wrote √12 as 2√3.
There are different ways to do the problem and it is wonderful that we end up with the same correct answer!
--------
I should have typed as follows:
(4 + √12)(4 +√12)
16 + 8√12 + 12
16 + 12 + 8√12
28 + 8√12
28 + 8√4 x √3
28 + 8 x 2 x √3
28 + 16√3
55.65
28+16sq.root3
And 20
16
why wasn't the square root of 3 multiplied by 8. only the 2 was multiplied by 8
2 sqrt of 3 is how many sqrts of 3 you have. Now if you multiply 2 sqrts of 3 by 8 how many sqrts of 3 will you have? 16 sqrts of 3. IOW, regard sqrt of 3 as a unit. 2 sqrt of 3 then is two of those units. Now, multiply those two units by 8. How many units do you have? 16 units. Now, turn those units back into sqrts of 3 and how many of them do you have? 16 sqrts of 3. I hope I made some kind of sense for you.
You’re good at simplifying but my heads still spinning
Perhaps you could structure your examples such that you can assume that your online students have worked through all the previous videos, so you wouldn't have to go back and re-explain basic math concepts in order to solve whatever given problem? Your voice is very soothing to those who are anxious about math, but you seem to have decided at the beginning of each video that you have to assume that your audience knows nothing about math, so you explain every math concept a student needs to understand in order to solve a problem. Perhaps you could number your videos, then refer the students back to the correct numbered video to learn *only* the concepts need to solve this problem?
If you were to do this, your explanations would be more concise and to the point, which would not only help the people who are trying to understand, but also you, in that you don't have to start every video with a ten minute explanation of all the math concepts the student would have to understand to solve a given problem. This is just a suggestion. Feel free to reject it out of hand. As an engineer who loves math, I really do enjoy your videos.
This can't be solved, it's not an equation...
Your math skills are better than your grammar skills(“anyways, finish out.”)
Shamefully simple.
Greetings. The answer is
4(7+4(3^1/2)).
(4+(12^1/2))^2,
(4)^2+2(4)(12^1/2)+(12^1/2)^2,
16+8(12^1/2)+12,
28+8(4^1/2)(3^1/2),
28+16(3^1/2),
4(7+4(3^1/2)).
Got lost in all of this!
This is a problem which can be solved in less than 20 seconds. He is making a lot of unnecessary talk which can possibly confuse a beginner.
Off the top of my head, the answer should be 28. The square root of 12 squared is 12 and 4 squared is 16.
Aaaaand ... I got ti wrong. LOL. :)
and your spelling too...
I can't solve it and I don't care because I will never use it practically in my life !!!
Nope didn't help a bit. Still do not know how to do this. I got lost in a maze of numbers!
Greetings. Try again. Don't give up. You can absolutely do it.
This can't be solved, it's not an equation...
True, there is only simplification possible.
@@panlomito evaluation, as it's only numbers...
An equation is defined as two expressions separated by an equals sign. An expression can have numbers, variables and mathematical operations, but it can also be a simple as a single number (a constant expression). The "?" in the given example is a symbol representing an unknown value ... in other words a variable. Based on those definitions, the given example is an equation which can be solved.
@paulfrank8738 wrong, an equation must have at least one unknown to be determined. The above is a numerical expression to be evaluated. Example, 7 + 9 is not an equation, but 7x + 9 = 12 is. Your math knowledge is lacking, you're not that smart. The ? Does not represent an unknown. 7 + 9 = 16 is not an equation because 7 + 9 does not equal 16 it's IDENTICALLY EQUAL to 16. In other words 7+9 is 16 written in a different form. The video asks to work out the value of a numerical expression.
@@rampakeshbharat1938Yup. the "?" is an unknown. Just because it isn't x or y doesn't mean it's not an unknown.
This can't be solved, it's not an equation...
Stop trying to confuse people who are trying to learn some Math. Everyone who is capable of googling "equation" knows you're wrong.
@@paulfrank8738 well clearly you don't as this is not an equation, it's a numerical expression to be evaluated...