Official Solution:
What is the value of \( \sqrt{25 + 10 \sqrt{6} } + \sqrt{ 25 - 10 \sqrt{6} }\) ?
A. \(2\sqrt{5}\)
B. \(\sqrt{55}\)
C. \(2\sqrt{15}\)
D. 50
E. 60
First, square the given expression to eliminate the square roots, but remember to take the square root of the result at the end to balance the operation and obtain the correct answer.
Important for the GMAT: \((x+y)^2=x^2+2xy+y^2\) and \((x-y)^2=x^2-2xy+y^2\).
Following these rules, we get:
\((\sqrt{25 + 10 \sqrt{6} } + \sqrt{ 25 - 10 \sqrt{6} })^2 =\)
\(=(\sqrt{25 + 10\sqrt{6} })^2+2(\sqrt{25 + 10\sqrt{6} })(\sqrt{25 - 10\sqrt{6} })+(\sqrt{25 - 10\sqrt{6} })^2=\)
\(=(25+10\sqrt{6})+2(\sqrt{25 + 10\sqrt{6} })(\sqrt{25 - 10\sqrt{6} })+(25-10\sqrt{6})\).
Note that the sum of the first and third terms simplifies to \((25+10\sqrt{6})+(25-10\sqrt{6})=50\), so we have:
\(50+2(\sqrt{25 + 10\sqrt{6} })(\sqrt{25 - 10\sqrt{6} })=\)
\(=50+2\sqrt{(25 + 10\sqrt{6})(25 - 10\sqrt{6}) }\).
Another
important concept for the GMAT: \((x+y)(x-y)=x^2-y^2\). Using this, we can simplify further:
\(50+2\sqrt{(25 + 10\sqrt{6})(25 - 10\sqrt{6})}=\)
\(=50+2\sqrt{25^2-(10\sqrt{6})^2)} = \)
\( = 50+2\sqrt{625-600}=\)
\(=50+2\sqrt{25}=\)
\(=60\).
Finally, remember to take the square root of this value to obtain the correct answer: \(\sqrt{60}=2\sqrt{15}\).
Answer: C