2003 AMC 10A Problem 9 — Solution | LIVE by Po-Shen Loh

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2003 AMC 10A Problem 9

Attempt Problem 9 of the 2003 AMC 10A 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 2003 AMC 10A solutions, or check the answer key.

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

9.

Simplify $\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}.$

$\sqrt{x}$

$\sqrt[3]{x^2}$

$\sqrt[27]{x^2}$

$\sqrt[54]{x}$

$\sqrt[81]{x^{80}}$

Answer: A

Concepts:radical exponent

Difficulty rating: 1350

Solution:

Working outward, $x\sqrt{x} = x^{3/2},$ and its cube root is $x^{1/2}.$

Then $x \cdot x^{1/2} = x^{3/2},$ whose cube root is again $x^{1/2}.$

Repeating once more, $x \cdot x^{1/2} = x^{3/2},$ whose cube root is $x^{1/2} = \sqrt{x}.$

Thus, the correct answer is A.

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