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Fill in the blanks with positive integers:
(3 + sqrt(5) - 1 + 2*sqrt(3) + 5)^2 = ___ + ___ *sqrt(5)

 Aug 22, 2024
 #1
avatar+1908 
+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! :)

 Aug 22, 2024
edited by NotThatSmart  Aug 22, 2024

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