dendenden wrote:
Quote:
Square the given expression to get rid of the roots, though don't forget to un-square the value you get at the end to balance this operation and obtain the right answer:
Hi, can anyone explain why we need to un-square the value? What is the rule or logic behind this?
Thank you.
Squaring an expression containing square roots is a common algebraic technique used to eliminate the square roots and simplify the expression. Conversely, taking the square root is an operation performed to balance the initial squaring. Essentially, first we square to simplify and eliminate the square roots but at the end to we get the squared value of the expression, and to get the original value, we need to unsquare.
Below is more detailed solution, which might help:
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