Simplify [√[3]√[6]a^9]^4[√[6]√[3]a^9]^4; the result is:

1956 AMC 12 Problem 9

Attempt Problem 9 of the 1956 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 1956 AMC 12 solutions, or check the answer key.

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

Simplify

\left[\sqrt[3]{\sqrt[6]{a^9}}\right]^4\left[\sqrt[6]{\sqrt[3]{a^9}}\right]^4;

the result is:

a^{16}

a^{12}

a^8

a^4

a^2

Replace each

n

th root by an exponent of

\frac{1}{n}

Each bracket has exponent

9\cdot\frac16\cdot\frac13\cdot4

For each of the two factors, the nested roots and outer fourth power multiply the exponent of

a

9\cdot\frac16\cdot\frac13\cdot4=2.

Thus each factor is

a^2,

and their product is

a^2a^2=a^4.

Therefore, the correct answer is D.