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Chapter 8: Problem 65

In simplifying \(36^{\frac{3}{2}},\) is it better to use \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) or \(a^{\frac{m}{n}}=(\sqrt[n]{a})^{m} ?\) Explain.

Short Answer

Expert verified
The second formula is preferable when simplifying \(36^{\frac{3}{2}},\) as it involves less complicated calculations. The simplified result is 216.

Step by step solution

01

Analysing Option 1

Here is the first formula: \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\). This formula involves exponentiation first, then taking the nth root. When applied to the exercise, we would calculate \(36^{3}=46656,\) and then take the square root of 46656, which might be a cumbersome calculation.
02

Analysing Option 2

Here is the second formula: \(a^{\frac{m}{n}}=(\sqrt[n]{a})^{m}\). This option requires to take the root first, and then exponentiate. If used, we would first calculate the square root of 36, which is 6, and then raise 6 to the 3rd power, yielding 216. The calculations included in this option seem a bit more manageable than the first.
03

Choosing the Best Option

Based on the analysis of steps required in each option, Option 2 seems to require less complicated operations, so it's preferable in this case.
04

Applying the Chosen Option

Applying the second formula, \(36^{\frac{3}{2}}=(\sqrt{36})^{3}\). The square root of 36 is 6, and 6 to the 3rd power is 216. Therefore, \(36^{\frac{3}{2}}=216.\)

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