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Chapter 33: Problem 953

Evaluate \(16^{-3 / 4}\)

Short Answer

Expert verified
The final expression for \(16^{-3/4}\) is \(\frac{1}{8}\).

Step by step solution

01

Rewrite the expression in radical notation

Recall that \(a^{m/n}\) can be written as \(\sqrt[n]{a^m}\). So, let's rewrite the given expression using this notation: \(16^{-3/4} = \sqrt[4]{16^{-3}}\)
02

Simplify the expression

Now, we know that \(16 = 2^4\), so we can replace the 16 in the expression with its equivalent power of 2: \(\sqrt[4]{16^{-3}} = \sqrt[4]{(2^4)^{-3}}\) Next, we apply the exponent rule \((a^m)^n = a^{mn}\): \(\sqrt[4]{(2^4)^{-3}} = \sqrt[4]{2^{-12}}\) Now, we simplify the expression inside the 4th root: \(\sqrt[4]{2^{-12}} = \frac{1}{\sqrt[4]{2^{12}}}\) Finally, we know that \(2^{12} = (2^3)^4 = 8^4\), which simplifies the expression: \(\frac{1}{\sqrt[4]{2^{12}}} = \frac{1}{\sqrt[4]{8^4}} = \frac{1}{8}\) So, \(16^{-3/4} = \frac{1}{8}\).

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