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Full Prep Academy
Приєднався 11 вер 2023
Welcome to Full Prep Academy, your go-to destination for straightforward and comprehensive math education. Here, we focus on creating, producing, editing, and publishing high-quality math videos tailored for all ages. No gimmicks, just a commitment to delivering clear and concise content that empowers learners at every level. Join us on the journey to mastery as we break down complex mathematical concepts in a way that makes learning both accessible and enjoyable. Welcome to a no-nonsense approach to math education at Full Prep Academy.
Is it possible to prove 2 = -2?
In this video, we explore a mathematical approach to show that 2=−22=−2 using basic arithmetic principles. We'll delve into the use of absolute values and exponent rules to demonstrate how seemingly contradictory statements can arise from arithmetic manipulations.
Here’s what we’ll cover:
Absolute Value Analysis: We’ll start by examining how absolute values work and how they can sometimes lead to unexpected results.
Exponent Rules: We’ll apply rules of exponents to see how they interact with the absolute value and influence the equality of numbers.
Step-by-Step Proof: We’ll guide you through each step of the arithmetic process to illustrate how we arrive at the equation 2=−22=−2 under specific conditions.
This video aims to provide a clear understanding of how these arithmetic rules can create scenarios that challenge our usual interpretations of equality.
Watch to see how mathematical properties can lead to intriguing conclusions and deepen your appreciation for arithmetic’s subtleties.
Here’s what we’ll cover:
Absolute Value Analysis: We’ll start by examining how absolute values work and how they can sometimes lead to unexpected results.
Exponent Rules: We’ll apply rules of exponents to see how they interact with the absolute value and influence the equality of numbers.
Step-by-Step Proof: We’ll guide you through each step of the arithmetic process to illustrate how we arrive at the equation 2=−22=−2 under specific conditions.
This video aims to provide a clear understanding of how these arithmetic rules can create scenarios that challenge our usual interpretations of equality.
Watch to see how mathematical properties can lead to intriguing conclusions and deepen your appreciation for arithmetic’s subtleties.
Переглядів: 22
Відео
Product of two consecutive integers
Переглядів 3422 місяці тому
I solve the problem: The product of two consecutive numbers is 210. What are those numbers? I demonstrate quadratics and how to solve quadratic equations by factoring. You actually end up with two pairs of numbers in the end. #math #quadratics
This is WRONG ...(in real numbers)
Переглядів 1,7 тис.2 місяці тому
Has your teacher ever said the square root gives two answers? This is wrong. If your teacher ever says this, please do correct them. The square root of a number is the principal root and only outputs non-negative numbers. #maths #algebra
Can you do this without calculator?
Переглядів 192 місяці тому
Can you solve this question without a calculator? The answer is absolutely, yes! You can use the difference of squares technique along with FOIL to get this question on any exam and test without the use of technology. All you need is your brain! #maths #arithmetic
Sum of three consecutive integers
Переглядів 2 тис.2 місяці тому
In this video, we show you how to deal with a common algebra problem where you have to total or sum three consecutive integers together. In this problem, you have to isolate for x and then determine what the middle integer is. By the end of the video, you should have a basic understanding of finding solutions to linear equations. #math #algebra
Do you know this math trick?
Переглядів 2432 місяці тому
Do you know this math trick? It's a little algebra technique to solve problems like this. This was taken from a math olympiad and it requires a little bit of thinking and a little bit of algebra to get the job done. It's not too bad, but knowing something about products and about prime numbers can lead you to a solid conclusion. It also starts with the premise that x and y are both positive num...
Asking AI for a Very Hard Trig Identity
Переглядів 368 місяців тому
Asking AI for a Very Hard Trig Identity
Complete Factoring Crash Course in 4 Minutes
Переглядів 1648 місяців тому
Complete Factoring Crash Course in 4 Minutes
Best Way to Prove Sum of Natural Numbers Formula
Переглядів 2269 місяців тому
Best Way to Prove Sum of Natural Numbers Formula
What is a Derivative? Best Explanation
Переглядів 29410 місяців тому
What is a Derivative? Best Explanation
My teacher was saying this to me all the time. That it is only positive it is. I was the one insisting on the negative's addition. On graphs too.
14
The answer is 14. 9 ÷ 3 = 3 3 × 3 = 9 9 + 5 = 14
B) 25. You would divide then multiply becauss MD are equal its just left to right :P
Can anyone explain why its not 25? Why wouldnt you do 25/5 and then multiply for 25? Am i missing something?
That IS what you're meant to do
@@doobimasada2487 yeah these comments keep making me feel stupid cause they are all saying "A(1)" 💀
1 A is the answer using bodmas rule
14
14
I normally write it out like linear factors and distribute. 52 = 50 + 2 52^2 = (50 + 2)(50 + 2) = 2500 + 100 + 100 + 4 = 2700 + 4 = 2,704.
The answer is 1. Change the ➗️ sign to a fraction line, then work the denominator separately. The expression becomes 25/5(5) = 25/25 = 1. On the other hand, if you did the multiplication before the division, you still get 25 ÷ 25 = 1.
You're saying you're doing the division first (as you should) but you are implying brackets around the 5×5 in order to put them on the denominator of your fraction. These brackets are not present in the term, it comes as 25÷5×5 and you do it in order left to right. You turned this into 25÷(5×5) which is changing the term entirely
Well... if you take this as a fraction, it is definitely 1, because 25/5×5 is 1. But if you add a multiplication sign between the 5 and the bracket, it could probably be like this: 25 ÷ 5 × 5 and you go 25 ÷ 5 is 5 and that × 5 is 25
You make it difficult. Just pass 2x2 tothe left side. -x2: -1,x2:1, x:-1,1
Awesome ❤❤
14
14
A1
1
As much as it would be funny to say both, it’s one
No.
69... interesting 🙂
Nice
helped alot croski no bizzy bap
UA-cam really out here recommending this to my British ass
Welcome!
Are there any rational numbers which can stand in for X and Y to make the first statement true? (3x-2y=6) Yes, I understand that the question is asking for the values of neither X nor Y.
Yes. We can translate this equation to function: 3x-2y=6 => 1.5x-3=y which has infinite solutions in rational numbers
@@Yaromir2008 Thank you. This has been driving me crazy. I had been trying to make sense of it, but I was coming at it from the wrong angle. It's been too long since I was in high-school, and this just doesn't come up in my daily life. (At least not in ways where I am tempted to express it as this kind of formula)
You can also replace x= (6+2y) /3 and solve 8^[ (6+2y) /3]/4^y. Result: 2^6.
me after 3sec: 168
0:51 but why?
its right but you don't need to to the denomenater because it dosen't make a diffrence :)
(-1±√211)/2 just by looking at the thumbnail
Ur wrong bro. Its -14,-15 and 14,15 just by looking at the thumbnail.
20
The clue for that kind of problem: where you have x with the power of 1 there is only one solusion ex. √x = 3 x = 9 where you have x with the power of 2 there is two solusions ex. x^2 = 9 x = 3 or x = -3
From line 4 you wrote +-x=4, and using line 2 you can write +-x=sqrt(16). 4 is a valid solution to x so therefore plugging it in reads +-4 = sqrt(16). I don't understand where I went wrong.
Yes, it is true that x² = 16 is equivalent to ±x = √16. But you need to understand that "±" hides that this is not one equation but a sort of system of two equations that are joined by an "OR": ±x = √16 really means +x = √16 OR -x = √16 You pointed out that 4 is a solution to the original equation x² = 16. In the system of equations this translates to +4 = √16 OR -4 = √16 Now keep in mind that it is not the case that both equations must simultaniously be true. This is because they are joined by an "OR": It suffices if one of them is true, then the whole system of equations evaluates to true. Now, because √16 is defined to be the value 4, the first equation is true and the second equation is false. Therefore, the whole system of equations evaluates to true which proves that 4 is indeed a solution to the equation. Note that -4 is also a solution. If you plug it into the system of equations, the first equation is now false while the second equation is true (as -(-4) = 4). This makes the whole system of equations true so that -4 is another solution to the original equation. Hope that helps 💪
@@complexcreations5309 Thanks for the response. It makes sense. But now the statement sqrt(16) = +-4 is actually sqrt(16) = +4 OR sqrt(16) = -4 which is a true statement even if sqrt(16) = +4 is only ever the true equivalence. I just point this out because the video claims it is wrong.
@@user-ze2yk7cd7g Well, this is a case of misusing notation. When you write ± you create a system of equations joined by OR. But this is not what people want to express by stating √16 = ±4. What it actually means is that if you create a system of equations by replacing ± with + or - respectively, at least one of the equations is true. But what people want to express by stating √16 = ±4 is different. What they mistakenly think is that √16 is equal to both 4 and -4 (i. e. creating a system of equations with AND). The issue here is that ±4 is not "the value 4 AND the value -4" but instead "the value 4 OR the value -4". So yes, technically √16 = ±4 is right under the aforementioned definition of ±, but because that is not how you typically read ± when it appears in an equation without a variable (which serves to clarify that you in fact deal with an equation which needs to be decomposed further) I think you should refrain from using ± in such a way as it will only lead to misunderstanding and confusion.
I don't fully comply with the video. He claims that the result of a principal square root only has one single value and is always an absolute number but he then mentions that the result of an absolute number still has two solutions. So due to transitivity it would indeed mean that in the end (an equation) of the square root has two solutions (while the outcome just of the square root still only provides a single value). Like after "x^2^1/2 = |x|" he further resolves to "|x| = |+/- x|". So "16^1/2" isn't just 4 its actually "|4|" and so it can be +4 as well as -4. If you have an equation like "a^2 = b^2" it resolves to "|a| = |b|" and so you have four possible solutions. |+a| = |+b| |+a| = |-b| |-a| = |+b| |-a| = |-b|
It's not "+4 = sqrt(16)" nor "-4 = sqrt(16)" it's "|4| = sqrt(16)". You can't alter the equation itself you can only show the possible solutions. So the complete equation would be "x = sqrt(16) = |4| = |+/- 4|", so there are two solutions with "x1 = +4" and "x2 = -4". Both are equally correct and if it would be a guessing game there is no way of knowing which of them is used. Basically such a behavior is used in cryptography where an input is put into an aquation which provides a single result like "-4^2" only becomes 16 but trying to reverse it gives you multiple solutions where you don't know what the original input was when you try to calculate it backwards. Like "x % 2" either resolves to 0 or 1 but where is no way in knowing what the original number was. You would only know if it was odd or even.
Thank you! I have a PhD in math, algebraic number theory, and I'm glad to see videos addressing this topic correctly. I like to offer to my students: When you introduce the square root to an equation, that's where you add the "±", but if the √ symbol is already there then the selection of + or - has already been made. I also will ask them, if the ± is needed (e.g., √16 = ±4) then "Why does the quadratic equation have a '±' in the numerator?" That would be redundant if 3+√16 simplifies down to 3+(±4).
What if your teacher marks the right answer as wrong?
Then show him the second and third paragraph of the Wikipedia article on square roots
Beautifully explained but for what practical purpose are we doing this?
20
In my head. 60 divide three is 20. The three consective numbers must be one less than 20 ,20 itself and 20 plus one. That is 19+20+21 = 60 easy peasy.
19,20, 21
0:01 a legend was born
If you make x the middle integer the answer just pops out. (x-1)+x+(x+1)=60 -> 3x=60 -> x = 20. But really the simplest way to solve this is to guess and check. Your first guess will probably be 20.
3x4x5=60
A product is not a total
Three consecutive numbers equals sixty. Sixty divided by three equals twenty. Twenty plus one is twenty one and twenty minus one is nineteen. Nineteen plus twenty plus twenty one is sixty. No algebra needed.
I got x=19 and worked it just like you . Misread question, need to look for middle number now! Thank you
Don’t need algebra to figure that one out.
These forms of eq^ns are called diophantine eq^ns and are especially more frequent in various olympiads. They have a pretty unique way of solving em and are pretty cool.
Simpler way would be to do x-1, x, x+1. That way the ones cancel out and you're solving directly for the middle integer
Wow. very nice. I hope you'll share more video with us.🎉
Beautiful algebra, but for such a simple problem, arithmetic is your friend
Trying myself first (disclaimer: not good at math lol) Edit: disclaimer 2, i make a mistake but correct for it later. Sorry for all the rambling. This turned out into writing down my thoughts as I had them xD but I think I learned a lot in the process! Now to watch the short xD 3x-y=12 Any value 3x produces is a multiple of 3 and because 12 is also a multiple of 3 y should be a multiple of 3 too or 0. Lets start then with y = 0 because thats easy :p so x = 4 But think about it y = (x - 4)3 If x = 7 and y = (7 -4) 3= 9 then 7×3 = 21 and 21 - 9= 12 I think that pretty much deals with the top equation so now to apply that to the second To start ill make the easy x=4 y=0 8⁴ = 64² = 16 + 240 + 240 + 3600 = 4096 2⁰= 2 (to the power of zero is the same as times 1 if I remember correctly) So 4096/2 = 2048 Now I did notice that is a power of 2 (4 8 16 32 64 128 256 512 1024 2048, so 2¹¹) and i also feel like there should probably be a different way to get to that without calculateing the actual values of 8⁴ and 2⁰ But first a quick check if 2⁰ hasn't messed things up for me. Using the conclusion of before ill change it to x =5 and y = (x-4)3 = 3 8⁵ = 4096 × 8 = 32768 2³ = 16 32768 / 16 = 2048 That's nice that worked out ^^" kind of feared i made a mistake but so far all good. I can say with certainty that the answer to the second is 2¹¹ I just want to improve my answer a bit by finding the "correct" way to devide powers (though this one still worked so I am happy already :p ) Now lets see if I remember how to devide powers. It has been a while. I am just going to try somethings that vaguely sound like something I might have learned at some point in the past lol. First lets try to devide the powers directly. So x = 4 and y = 0 so 4/0 and.... maybe not that one lol. What about x = 5 y = 3. I just realised a problem with this. Because of how I calculate y it can be bigger and smaller than x. X 5 = y 3 and x 10 = y 18. So this wont work. How else do I do it? Maybe work back from the answer. It is 2048 but also 2¹¹ and because we also have a power of 2 in our equation I feel like we are onto something there. Also looking at the 8 i think it may help to make it also 2 somehow. (Or make the 2 8 but because our answer is a multiple of 2 ill try 8 to 2 first) 8 to the power of x = 2 to the power of x + 3 because 2³ is 8. I think... double check 8⁴ = 4096 and 2⁷ is... way less because we keep multiplying by 2. So we have to raise them all to 8 so it should be to the power of x3? 4*3 = 12 and 2¹² is 4096 :D We got somewhere! So rather then 8⁴/2⁰ we have 2¹² / 2⁰ PANIC! I decided to dubbel check 2⁰ and it very much is not 2. I made a mistake. It is 1. You don't multiply by one it is 1. Oh dang. Fortunately the x5 y3 had the same result but this makes me worry a bit. Just to double check before moving on 3 × 4 - 1 = 11. So ehm, there is no way y = 1 is a thing. Well zeroes are weird anyway ^^" lets just stay at y= 3 and x =5. 3 × 5 = 15, 15 - 3 = 12. Thats good. And we already checked the second one with these numbers ( if someone knows how to reconcile y = 0 let me know...) Back to the regular programming. So now with 8⁵ is the same as 2¹⁵. So 2¹⁵/ 2³ = 2¹¹... is it? No wait... how did mess that up. Looking back I somehow thought 2³ was 16... my thought was the powers of 2 are 4 8 16. 16 being the 3rd. But it is not of course. It is the 4th! 4 is the second power. I sure hope I don't have any further mistakes... glad I wanted to look for a different solution or I would have missed it (my math exams took a long time xD). So the actual answer is 2¹²! And that is nice because that makes the calculation easier i think. Because now we have 2¹⁵ / 2³ = 2¹². Or 15 - 3 = 12! Oh and that also solves y = 0! Because 8⁴= 2¹² and 12 -0 = 12. How ever that is a bit of a cheat because 2⁰ is still 1 and 2 × 1 is 2 so it is 2¹²/ 2 which is 2¹¹. Frick. What about x= 6? That means y = (6-4)3 = 6. So 8⁶ = 2¹⁸ and 2¹⁸/2⁶ = 2¹²? I think that is right. It's a bit annoying though that y = 0 doesn't work. But every number created by an x higher then 4 should work. Any feedback on this is more then welcome. Would really appreciate it. Like i said at the start I am not very good at math but I love to learn so please share anything you want :3
Nice short! It is heartening to see the solution I came with subtracting the exponents to also being used here. But that does leave me with a question. If x = 4 and y = 0. Then this doesn't work right? 3 × 4 - 0 = 12 But 8⁴/2⁰ .... it does work. My bad. See the comment above for my confusion in whole xD For some reason i thought because 2⁰ = 1 i still had to multiply that 1 times 2. But i don't. So the 8⁴/2⁰ = 8⁴/1 = 8⁴ = 2¹² Well at least my brain was busy today xD thanks for this short!
2^12
N!=Nx(N-1)x...x2x1 0!=0x(0-1)x...x2x1 0!=0 Why this contradiction?
Formula only applies for N = 1
@@fullprepacademy Yes, but you can apply this formula for any N>=1, but why not for 0?
It is because a factorial is defined as a positive number ( N > 0 ) multiplied by one lower than it until you get to one. A consequence of this is that 0! = 1 can be demonstrated when N = 1. However, N cannot be less than 1 in the formula due to the definition.