Problem 89 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(\frac{1}{32^{\frac{4}{5}}}\).

Step 2: 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 \(\frac{1}{\sqrt[5]{32^4}}\).

Step 3: Evaluate the Exponential Function

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

Step 4: 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 \(\frac{1}{16}\).

Step 5: Final Solution

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