What is the value of square root{25 + 10square root(6)} + square root{25 - 10square root(6)} GMAT Problem Solving

Question: What is the value of \(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}\) ?

1. 2\(\sqrt5\)​
2. \(\sqrt{55}\)
3. 2\(\sqrt{15}\)
4. 50
5. 60

Answer:

Approach Solution 1:

We know that:

\((x+y)^2=x^2+y^2+2xy\)

\((x-y)^2=x^2+y^2-2xy\)

So we get:

\([\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}]^2=[\sqrt{25+10\sqrt{6}}]^2+[\sqrt{25-10\sqrt{6}}]^2+2[\sqrt{25+10\sqrt{6}}][\sqrt{25-10\sqrt{6}}]\)

\(=25+10\sqrt6+25-10\sqrt6+2[\sqrt{25+10\sqrt{6}}][\sqrt{25-10\sqrt{6}}]\)

Note that sum of the first and the third terms simplifies to \([\sqrt{25+10\sqrt{6}}][\sqrt{25-10\sqrt{6}}]\)=50, so we have \(50+2[\sqrt{25+10\sqrt{6}}][\sqrt{25-10\sqrt{6}}]\)therefore:

\(50+2[\sqrt{25+10\sqrt{6}}][\sqrt{25-10\sqrt{6}}]=50+2[\sqrt{25+10\sqrt{6}}][\sqrt{25-10\sqrt{6}}]\)

We also know that:

\((x+y)(x-y)=x^2-y^2,thus\)

\(50+2[\sqrt{25+10\sqrt{6}}][\sqrt{25-10\sqrt{6}}]=50+2\sqrt{25^2-(10\sqrt{6})^2}\)

\(=50+2\sqrt{625-600}=50+2\sqrt{25}=60\)

Recall that we should un-square these values to get the right answer: \(\sqrt{60}=2\sqrt{15}\)

Correct option: C

“What is the value of\(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}\)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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