Square Hint Squared

Algebra Level 1

Evaluate $\frac{ 63^4 - 37^4 } { 63^2 + 37^2 }.$

Hint: Use the difference of two squares identity .

The answer is 2600.

3 solutions

Brian Moehring
Feb 1, 2017

$\frac{63^4-37^4}{63^2+37^2} = \frac{(63^2+37^2)(63^2-37^2)}{(63^2+37^2)} = 63^2 - 37^2 = (63+37)(63-37) = (100)(26) = 2600$

Moderator note:

This solution makes very nice use of the fact that a technique can be repeated in the same problem -- here by applying difference of squares more than once, even with the expression $63^2 - 37^2$ which some might stop at but lends itself to the simple expression $(100)(26).$

Even when a calculation involves pure numbers, recognizing algebraic patterns can be useful!

Hint^2 :)

Good job

Chung Kevin - 4 years, 4 months ago

Yup, the steps becomes abundantly clear once we've identified that it's just a (repeated) difference of squares.

In this case, the expression is in the form of (a^4-b^4)/(a^2 + b^2), the numerator factors to (a^2-b^2)(a^2 + b^2)

And we're left with (a^2 - b^2), which can be further broken down to (a-b)(a+b)

Pi Han Goh - 4 years, 4 months ago

Where did I go wrong? Squaring everything once should leave me with 63^2 - 37^2 /63+37 -> 63^2 - 37^2 /100. Squaring everything again leaves me with 63 - 37/10 = 2.6

Shouldn't I multiply it by 10^4? It gives me a result of 26000. Where is my mistake?

Gabriel Souza - 4 years, 1 month ago

You can't just remove the squares like that. $(a^2 - b^2)/(c^2 - d^2)$ is not equal to $(a-b)/(c-d)$ .

Pi Han Goh - 4 years, 1 month ago

we should make it easy 63 * 63=3609 * 63=10287+8997=5542 * 544/665 answer 2600

Yassin Ali Elkhateb - 3 years, 11 months ago

I don't understand your sequences of equations.

The point of the question is that we don't have to evaluate $63^2$ .

Chung Kevin - 3 years, 11 months ago

A Former Brilliant Member
Feb 1, 2017

Nice use of colors :) We can use the difference of two squares identity once again for $63^2 - 37^2$ to further simplify the calculations.

$63 - 37 = 26$ , and $63 + 37 = 100$ . It is easy to multiply 26 with 100 to get 2600.

Pranshu Gaba - 4 years, 4 months ago

cool - how'd you make that image, please?

Terry Smith - 4 years, 4 months ago

Terry, we can use LaTeX on this site to render math

Agnishom Chattopadhyay - 4 years, 4 months ago

Yes and I've done that, but this is a graphic image.

Terry Smith - 4 years, 3 months ago

Read this

Pi Han Goh - 4 years, 4 months ago

A Former Brilliant Member
Jan 17, 2018

$\dfrac{63^4-37^4}{63^2+37^2}=\dfrac{(63^2-37^2)(63^2+37^2)}{63^2+37^2}=63^2-37^2=(63-37)(63+37)=26(100)=\boxed{2600}$

Square Hint Squared

The answer is 2600.

3 solutions

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