Fill in the blanks with positive integers:

(3 + sqrt(5) - 1 + 2*sqrt(3) + 5)^2 = ___ + ___ *sqrt(5)

LiIIiam0216 Aug 22, 2024

#1**+1 **

The following is using brute force because I sux at math. ;)

First, let's combine some like terms within the parenthesis. We have

\(\left(2\sqrt{3}+\sqrt{5}+7\right)^{2}\)

Since we can't simplify this forwards, let's set up the squaring process. Expanding the square, we get

\(\left(2\sqrt{3}+\sqrt{5}+7\right)\left(2\sqrt{3}+\sqrt{5}+7\right)\)

Now, let's use distributive property. Distributing every number in, and simplifying ALOT, we get

\({2\sqrt{3}\cdot \left(2\sqrt{3}+\sqrt{5}+7\right)+\sqrt{5}\cdot \left(2\sqrt{3}+\sqrt{5}+7\right)+7\left(2\sqrt{3}+\sqrt{5}+7\right)}\\\)

I'm not going to show the entire process, but I think you got it!

Now, remember that a square being square becomes an integer.

Distributing every single number in and doing some simplifying, we get

\(2\sqrt{15}+14\sqrt{3}+12+2\sqrt{15}+7\sqrt{5}+5+{14\sqrt{3}+7\sqrt{5}+49}\)

\(4\sqrt{15}+28\sqrt{3}+14\sqrt{5}+66\)

This is the furthest we can simplify, so it's our final answer.

Thanks! :)

NotThatSmart Aug 22, 2024