1986 AHSME Problems/Problem 5

Problem

Simplify $\left(\sqrt[6]{27} - \sqrt{6 \frac{3}{4} }\right)^2$

$\textbf{(A)}\ \frac{3}{4} \qquad \textbf{(B)}\ \frac{\sqrt 3}{2} \qquad \textbf{(C)}\ \frac{3\sqrt 3}{4}\qquad \textbf{(D)}\ \frac{3}{2}\qquad \textbf{(E)}\ \frac{3\sqrt 3}{2}$

Solution

We have $\sqrt[6]{27} = (3^{3})^(\frac{1}{6}) = 3^{\frac{1}{2}} = \sqrt{3}$ and $\sqrt{6 \frac{3}{4}} = \sqrt{\frac{27}{4}} = \frac{3 \sqrt{3}}{2}$ , so the answer is $(\sqrt{3} - \frac{3 \sqrt{3}}{2})^{2} = (-\frac{\sqrt{3}}{2})^{2} = \frac{3}{4}$ , which is $\boxed{A}$ .

1986 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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1986 AHSME Problems/Problem 5

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