2008 AMC 10B Problem 3 — Solution | LIVE by Po-Shen Loh

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2008 AMC 10B Problem 3

Attempt Problem 3 of the 2008 AMC 10B 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 2008 AMC 10B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

3.

Assume that $x$ is a positive real number. Which is equivalent to $\sqrt[3]{x\sqrt{x}}\,?$

$x^{1/6}$

$x^{1/4}$

$x^{3/8}$

$x^{1/2}$

$x$

Answer: D

Concepts:exponent radical

Difficulty rating: 940

Solution:

Since $\sqrt{x}=x^{1/2},$ we have $x\sqrt{x}=x^{1}\cdot x^{1/2}=x^{3/2}.$

Taking the cube root multiplies the exponent by $\tfrac13,$ giving $\left(x^{3/2}\right)^{1/3}=x^{1/2}.$

Thus, the correct answer is D.

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