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Chapter 10: Problem 39

Evaluate each exponential. $$ \left(\frac{16}{81}\right)^{-3 / 4} $$

Short Answer

Expert verified
\( \frac{27}{8} \text{ or } 3.375 \text{ (approx.)} \)

Step by step solution

01

Recognize the base components

Rewrite the fraction \(\frac{16}{81}\) in terms of exponentiations. Both 16 and 81 are powers of 2 and 3 respectively: \(\frac{16}{81} = \frac{2^4}{3^4}\).
02

Apply the negative exponent

Use the rule \(\frac{a}{b}^{-n} = \frac{b^n}{a^n}\) to get rid of the negative exponent: \(\frac{2^4}{3^4}\bigg)^{-3/4} = \frac{3^{4 \times 3/4}}{2^{4 \times 3/4}})\).
03

Simplify the exponents

Simplify the exponents inside the fraction: \(\frac{3^{4 \times 3/4}}{2^{4 \times 3/4}} = \frac{3^3}{2^3}\).
04

Evaluate the final expression

Calculate the values in the fraction: \(\frac{3^3}{2^3} = \frac{27}{8}\).

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Key Concepts

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

negative exponents
Negative exponents might seem tricky at first, but they follow a simple rule. Whenever you have a negative exponent, it essentially means you're dealing with the reciprocal of the number raised to the positive exponent. For example, if you have \(a^{-n}\), that's equivalent to \(\frac{1}{a^n}\). So, for the given exercise, \(\left(\frac{16}{81}\right)^{-3/4}\), we apply this rule to change the negative exponent: \(\left(\frac{16}{81}\right)^{-3/4} = \left(\frac{81}{16}\right)^{3/4}\). This makes solving the problem easier because now we don't have to deal with the negative exponent.
simplifying fractions
Simplifying fractions plays an essential role in solving exponent problems with fractions. In the provided exercise, we have \(\frac{16}{81}\). Both 16 and 81 are powers of integers: \(16 = 2^4\) and \(81 = 3^4\), allowing us to rewrite the fraction as \(\frac{2^4}{3^4}\). This rewriting process simplifies our calculations when we apply other exponent rules. When fractions are expressed as powers of integers, it becomes easier to apply exponent rules.
rule of exponents
Understanding the rule of exponents is key to solving problems like this. Here are integral rules to remember:
  • \(a^m \cdot a^n = a^{m+n}\)
  • \(\left(a^m\right)^n = a^{m\cdot n}\)
  • \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)
In our exercise, we use the rule \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). Looking into the provided step: \(\left(\frac{2^4}{3^4}\right)^{3/4} = \frac{2^{4\cdot 3/4}}{3^{4\cdot 3/4}} = \frac{2^3}{3^3}\), ensuring the exponents are distributed correctly across the fraction.
evaluating powers
The final step to solving the exponential expression is to evaluate the powers. In our exercise, we reached \(\frac{3^3}{2^3}\). To evaluate powers, simply compute the base raised to the exponent.

For 3鲁, it鈥檚 calculated as: 3 * 3 * 3 = 27.
For 2鲁, it鈥檚 calculated as: 2 * 2 * 2 = 8.
This gives us the final simplified answer: \(\frac{27}{8}\). Evaluating powers is a step by step multiplication process of the given base.

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Most popular questions from this chapter

A Panasonic Smart Viera E50 LCD HDTV has a rectangular screen with a 36.5 -in. width. Its height is 20.8 in. What is the length of the diagonal of the screen to the nearest tenth of an inch? (Data from measurements of the author's television.)

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Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt{\sqrt[3]{\sqrt[4]{x}}} $$
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