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Chapter 7: Problem 38

Evaluate each exponential. $$ \left(\frac{64}{125}\right)^{-2 / 3} $$

Short Answer

Expert verified
\( \frac{25}{16} \)

Step by step solution

01

Understand the Exponent

The given expression is \(\frac{64}{125}\) raised to the power of \(-\frac{2}{3}\). The negative exponent means we will take the reciprocal of the base, and the fraction exponent indicates taking both a root and a power.
02

Take the Reciprocal

Since we have a negative exponent, take the reciprocal of \(\frac{64}{125}\). So, \(\frac{64}{125}\) becomes \(\frac{125}{64}\). Now the expression is \(\frac{125}{64}\) raised to the power of \(\frac{2}{3}\).
03

Apply the Fractional Exponent

The exponent \(\frac{2}{3}\) indicates taking the cube root first, then squaring the result. Calculate the cube root of both the numerator and the denominator: \(\frac{\root 3 \big(125\big)}{\root 3 \big(64\big)} = \frac{5}{4}\).
04

Square the Result

Now that you have \(\frac{5}{4}\), square both the numerator and the denominator: \(\frac{5^2}{4^2} = \frac{25}{16}\). This is the final result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Exponent
When dealing with a negative exponent, it means we need to take the reciprocal of the base before applying the positive exponent. In our exercise, the base is \(\frac{64}{125}\). Taking the reciprocal swaps the numerator and the denominator. Thus, \(\frac{64}{125}\) becomes \(\frac{125}{64}\). Just think of a negative exponent as flipping the fraction. This step transforms our problem into a form where we can more readily apply further exponent rules.
Negative exponents may seem confusing at first, but they have a straightforward rule:
  • \( a^{-b} = \frac{1}{a^b} \)
Remember, taking the reciprocal is the first step, and is crucial before moving ahead with other operations.
Fractional Exponent
Fractional exponents indicate roots and powers at the same time. A fraction like \(\frac{2}{3}\) tells us to perform two operations: taking the cube root and then squaring the result. For example, in \((\frac{125}{64})^{\frac{2}{3}}\), the steps include: first taking the cube root of both the numerator and the denominator, and then squaring the results.
The reasoning here is:
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