Simplify (√[6]27-√634)^2.

1986 AMC 12 Problem 5

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Simplify

\left(\sqrt[6]{27}-\sqrt{6\frac34}\right)^2.

\frac34

\frac{\sqrt3}{2}

\frac{3\sqrt3}{4}

\frac32

\frac{3\sqrt3}{2}

Rewrite

27

as a power of

3

and

6\frac34

as an improper fraction

Simplify both radicals before squaring their difference

We have

\sqrt[6]{27}=(3^3)^{\frac{1}{6}}=\sqrt3

and

\sqrt{6\frac34}=\sqrt{\frac{27}{4}}=\frac{3\sqrt3}{2}.

Hence the expression is

\begin{aligned} \left(\sqrt3-\frac{3\sqrt3}{2}\right)^2 &=\left(-\frac{\sqrt3}{2}\right)^2\\ &=\frac34. \end{aligned}

Thus the correct answer is A.