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Chapter 3: Problem 6

Evaluate the indicated quantities. Do not use a calculator because otherwise you will not gain the understanding that these exercises should help you attain. $$ 8^{-5 / 3} $$

Short Answer

Expert verified
\(8^{-5/3}\) can be rewritten as \((\frac{1}{8})^{5/3}\). By simplifying the base, we get \((\frac{1}{2^3})^{5/3}\). Applying the power of a power property, we obtain \(\frac{1}{2^{3 \cdot (5/3)}}\), which simplifies to \(\frac{1}{2^5}\). Therefore, the final answer is \(8^{-5/3} = \frac{1}{32}\).

Step by step solution

01

Remove the negative exponent

According to the properties of exponents, a negative exponent can be removed by taking the reciprocal of the base. Therefore: \(8^{-5/3} = (\frac{1}{8})^{5/3}\)
02

Simplify the base

Notice that 8 can be written as a power of 2, so let's rewrite the base in terms of 2: \(\frac{1}{8} = \frac{1}{2^3}\) Now, substitute this back into the original expression: \((\frac{1}{2^3})^{5/3}\)
03

Evaluate the exponent

Now, apply the power of a power property by multiplying the exponents: \((\frac{1}{2^3})^{5/3} = \frac{1}{2^{3 \cdot (5/3)}}\) Simplify the exponent by multiplying 3 by the fraction (5/3): \(\frac{1}{2^{3 \cdot (5/3)}} = \frac{1}{2^5}\)
04

Final answer

Now that we have simplified the expression, we obtain the final answer: \(8^{-5/3} = \frac{1}{2^5} = \frac{1}{32}\)

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