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Chapter 0: Problem 89

Evaluate each expression without using a calculator. $$32^{-\frac{4}{5}}$$

Short Answer

Expert verified
The result of the expression \(32^{-\frac{4}{5}}\) is \(\frac{1}{16}\).

Step by step solution

01

Negative Exponent Rule

The first step is to follow the negative exponent rule, which states that \(a^{-n} = \frac{1}{a^n}\). Applying this, the expression will look like \(\frac{1}{32^{\frac{4}{5}}}\).
02

Fraction Exponent Rule

The second step is to follow the fractional exponent rule which states \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\). Applying this, the expression turns into \(\frac{1}{\sqrt[5]{32^4}}\).
03

Evaluate the Exponential Function

Next step is to evaluate the exponential function \(32^4\). Which results in 1048576.
04

Evaluate the Fifth Root

The fourth step is to evaluate the fifth root of \(1048576\). The fifth root of \(1048576\) is \(2^4 = 16\). So, the expression becomes \(\frac{1}{16}\).
05

Final Solution

After simplifying the fraction, the final result of the expression \(32^{-\frac{4}{5}}\) is \(\frac{1}{16}\).

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Most popular questions from this chapter

Factor completely. $$ x^{2 n}+6 x^{n}+8 $$

Simplify each expression. Assume that all variables represent positive numbers. $$ \left(8 x^{-6} y^{5}\right)^{\frac{1}{3}}\left(x^{\frac{5}{6}} y^{-\frac{1}{3}}\right)^{6} $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) toproduce a true statement. $$\left(4 \times 10^{3}\right)+\left(3 \times 10^{2}\right)=4.3 \times 10^{3}$$

Evaluate each expression without using a calculator. $$8^{\frac{2}{3}}$$

Exercises \(142-144\) will help you prepare for the material covered in the next section. Use the distributive property to multiply: $$2 x^{4}\left(8 x^{4}+3 x\right)$$
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