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If A=10+3-10-3 10-1, B=3+22, What is A - B?

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by Max@Math Revolution » Mon Mar 02, 2020 1:11 am

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[GMAT math practice question]

If \(A=\frac{\sqrt{\sqrt{10}+3}-\sqrt{\sqrt{10}-3}}{\sqrt{\sqrt{10}-1}}\) , \(B=\sqrt{3+2\sqrt{2}}\) , What is A - B?

A. -2
B. -1
C. 0
D. 1
E. 2
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Re: If A=10+3-10-3 10-1, B=3+22, What is A - B?

by Max@Math Revolution » Wed Mar 04, 2020 12:16 am
=>
\(A^2=\left(\frac{\sqrt{\sqrt{10}+3}-\sqrt{\sqrt{10}-3}}{\sqrt{\sqrt{10}-1}}\right)^2\)
\(A^2=\left(\frac{\sqrt{\sqrt{10}+3}-\sqrt{\sqrt{10}-3}}{\sqrt{\sqrt{10}-1}}\right)\left(\frac{\sqrt{\sqrt{10}+3}-\sqrt{\sqrt{10}-3}}{\sqrt{\sqrt{10}-1}}\right)\)
\(A^2=\frac{\sqrt{10}+3+\sqrt{10}-3-2\sqrt{10-9}}{\sqrt{10}-1}\)
\(A^2=\frac{2\left(\sqrt{10}-1\right)}{\sqrt{10}-1}\)
\(A^2=2\)

Thus we have A = √2.
\(B=\sqrt{3+2\sqrt{2}}=\sqrt{\left(2+1\right)-2\sqrt{2\cdot1}}=\sqrt{2}+1\)
Then A – B = √2 – (√2 + 1) = -1.

Therefore, B is the answer.
Answer: B
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