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If [ (a⁹)^(1/6) ]^(1/3)⁴ * [ (a⁹)^(1/3) ]^(1/6)⁴ = a^k, that is, { [ (a⁹)^(1/6) ]^(1/3) }⁴ * { [ (a⁹)^(1/3) ]^(1/6) }⁴ = a^k, find the value of k.

1. 2
2. 4
3. 6
4. 8

Correct answer: 4

Solution

First factor: start with a⁹, take 6th root -> a^(9/6) = a^(3/2). Take cube root -> a^(3/2 * 1/3) = a^(1/2). Raise to power 4 -> a^(4/2) = a². Second factor: start with a⁹, take cube root -> a^(9/3) = a³. Take 6th root -> a^(3/6) = a^(1/2). Raise to power 4 -> a². Product: a² * a² = a⁴. Therefore k = 4.

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