Which of these four numbers, √π^2, √[3]0.8, √[4]0.00016, √[3]-1×√(0.09)^-1, is (are) rational?

1958 AMC 12 Problem 8

Attempt Problem 8 of the 1958 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1958 AMC 12 solutions, or check the answer key.

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Which of these four numbers,

\sqrt{\pi^2},

\sqrt[3]{0.8},

\sqrt[4]{0.00016},

\sqrt[3]{-1}\cdot\sqrt{(0.09)^{-1}},

is (are) rational?

none

all

the first and fourth

only the fourth

only the first

Evaluate each radical separately and remember that

\pi

is irrational

The last expression uses

\sqrt[3]{-1}=-1

and

(0.09)^{-1}=\frac{100}{9}

The first number is

\pi,

which is irrational. The cube root of

\frac{4}{5}

is irrational. Also

\sqrt[4]{0.00016} =\sqrt[4]{16\cdot10^{-5}}

is irrational. The fourth number is

(-1)\sqrt{\frac{100}{9}}=-\frac{10}{3},

which is rational.

Therefore, only the fourth is rational, and the correct answer is D.