/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 59 Evaluate each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 59

Evaluate each expression. $$ \left(\frac{4}{5}\right)^{-2} $$

Short Answer

Expert verified
The expression evaluates to \( \frac{25}{16} \).

Step by step solution

01

Identify the Negative Exponent

The expression \(\frac{4}{5}\right)^{-2} \) has a base of \( \frac{4}{5} \) and an exponent of -2. The negative exponent indicates that the base should be inverted (reciprocal).
02

Take the Reciprocal

Taking the reciprocal of \( \frac{4}{5} \) means inverting the fraction. Therefore, \( \frac{4}{5} \) becomes \( \frac{5}{4} \).
03

Apply the Positive Exponent

Now apply the positive version of the exponent to the reciprocal. Calculate \( \frac{5}{4} \) raised to the power of 2: \(\frac{5}{4}\right)^2 \). This means multiplying \( \frac{5}{4} \) by itself.
04

Perform the Multiplication

Multiply \( \frac{5}{4} \) by \( \frac{5}{4} \): \(\frac{5}{4} \times \frac{5}{4} = \frac{25}{16}\right) \).

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Key Concepts

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

Reciprocal
When dealing with negative exponents, one of the key concepts is the reciprocal. The reciprocal of a number is simply one divided by that number. For example, the reciprocal of 2 is \( \frac{1}{2} \). If a number is already a fraction, such as \( \frac{4}{5} \), then its reciprocal is found by flipping the numerator and the denominator, making it \( \frac{5}{4} \). This is important because a negative exponent essentially tells us to take the reciprocal of the base and then apply the positive exponent to that reciprocal.
Fraction Multiplication
Fraction multiplication is straightforward once you understand the method. When multiplying two fractions, you just multiply the numerators together and the denominators together. For example, when multiplying \( \frac{5}{4} \) by \( \frac{5}{4} \), you multiply the numerators (5 and 5) together to get 25, and the denominators (4 and 4) to get 16. So, \( \frac{5}{4} \times \frac{5}{4} = \frac{25}{16} \). This is useful when working with expressions that involve raising fractions to a power, as we'll see in our example problem.
Positive Exponents
Positive exponents are more familiar and straightforward to handle. In the context of our example problem, once the base fraction has been inverted to its reciprocal, the negative exponent becomes positive. This means we apply the exponent in the usual way. For instance, raising \( \frac{5}{4} \) to the power of 2 involves multiplying \( \frac{5}{4} \) by itself: \( \left(\frac{5}{4}\right)^2 = \frac{5}{4} \times \frac{5}{4} = \frac{25}{16} \). Positive exponents indicate repeated multiplication, which can be easy to manage once you understand fraction multiplication.

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Most popular questions from this chapter

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \left(3 p^{-4}\right)^{2}\left(p^{3}\right)^{-1} $$

Find each product. $$ (3 r+2 s)\left(r^{3}+2 r^{2} s-r s^{2}+2 s^{3}\right) $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(\frac{p^{2} q^{-1}}{2 p^{-2}}\right)^{2} \cdot\left(\frac{p^{3} \cdot 4 q^{-2}}{3 q^{-5}}\right)^{-1} \cdot\left(\frac{p q^{-5}}{q^{-2}}\right)^{3} $$

Two expressions are given. Replace x with 3 and y with 4 to show that, in general, the two expressions are not equivalent. $$ (x+y)^{5} ; \quad x^{5}+y^{5} $$

Write each number in scientific notation. $$ 0.875 $$
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