Official Guide Explanation:
Problem Solving #11

Background

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Solution and Metadata

Question: 11
Page: 153
Difficulty: 5 (Moderate)
Category 1: Arithmetic > Powers and Roots of Numbers > Roots other than 2
Category 2: Arithmetic > Powers and Roots of Numbers > Roots

Explanation: When dealing with very small decimals, like 0.000064, it is often easier to rewrite them using scientific notation. 0.000064 = 64 * 10^-6, so we can think of the question as rt[ ³rt[64 * 10^-6]]. Start from the inside, and to make things simpler, think of 64 as 2⁶. Thus, the first step is simplifying ³rt[2⁶ * 10^-6], which is equivalent to (2⁶ * 10^-6)^1/3. Distributing the exponent, we get (2⁶)^1/3 * (10^-6)^1/3, or 2² * 10^-2. The next step is keep working our way out of the radical signs: rt[2² * 10^-2] = (2²)^1/2 * (10^-2)^1/2 = 2 * 10^-1 = 0.2, which is choice (E).

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