Need Help- Addition and Subtraction under Square Roots

Question

Hello

I am having difficulty with how to simplify addition and subtraction under a square root. Below is the problem and how it is stated in the Foundations of Math Manhattan Prep book. I also attached a copy of me trying to work out the problem. (AJ-WORK1 AND AJ-WORK2) The other attachment is the solution info that the book provides which isn't that clear to me. If someone can explain what I am doing wrong or how to look at this problem in a different way as well as some other examples perhaps ; that would be really great.

QUESTION #56 from Chapter 3 in FOM 
Simplify the following root:Express expressions as integers

√(36^2 +15^2)

Thanks in advance

Felling super frustrated-AJ

akela · Answer

Hi  Lnstnbass4 ,

I guess you made a mistake when you taking  3^2  out.

\sqrt{36^2+15^2} = \sqrt{(2^2*3^2)^2+(3*5)^2} = \sqrt{2^4*3^4+3^2*5^2} = \sqrt{3^2(2^4*3^2+5^2)} = \sqrt{3^2(16*9+25)} = \sqrt{3^2(144+25)} = \sqrt{3^2*169} = \sqrt{3^2*13^2} =3*13=39

Hope this helps!

ccooley · Answer

There's a quick way to solve this problem. (That's true on a lot of GMAT problems where the numbers seem too big to work with. They know you don't have a calculator, and they aren't really interested in testing your arithmetic skills beyond the basics, so if it looks like you might have to do a ton of arithmetic to solve a problem, there's probably another way to do it.)

Here's the right thought process for this problem.

1. There isn't really a math rule for adding together exponents, like 36^2 + 15^2.
2. However, there are two situations where you might see something like this on the test.
3. One is where you have two exponents with the same base added together, like 84^2 + 84^3. When you see this, factor out a common term. For instance, 84^2 + 84^3 = 84^2(1 + 84). 
4. Another is where you're using the Pythagorean Theorem on a right triangle. That's a^2 + b^2 = c^2, or √(a^2 + b^2) = c. 
5. Does this problem look like one of those two situations? It doesn't look like the first one, but it  could  look like the second one. Are 15 and 36 two sides of a right triangle? 
6. A ratio of 15 to 36 is the same as a ratio of 15/3 to 36/3, or 5 to 12. There's a special right triangle that has sides with a ratio of 5 to 12! It's a (5,12,13) triangle.
7. Therefore, (5*3)^2 + (12*3)^2 = (13*3)^2. 
8. Rephrasing that: sqrt(15^2 + 36^2) = sqrt(39^2) = 39.

Need Help- Addition and Subtraction under Square Roots

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