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Chapter 5: Problem 12

Simplify each numerical expression. \(\frac{1}{\left(\frac{4}{5}\right)^{-2}}\)

Short Answer

Expert verified
The simplified expression is \( \frac{16}{25} \).

Step by step solution

01

Understand the Negative Exponent

The negative exponent \(\left(\frac{4}{5}\right)^{-2}\) means you take the reciprocal of \(\frac{4}{5}\) and then raise it to the positive of that exponent. So, \(\left(\frac{4}{5}\right)^{-2}\) is equivalent to \(\left(\frac{5}{4}\right)^{2}\).
02

Take the Reciprocal

Switch the numerator and denominator of \(\frac{4}{5}\) to get the reciprocal, which is \(\frac{5}{4}\).
03

Apply the Positive Exponent

Now raise \(\frac{5}{4}\) to the power of 2. This means multiplying \(\frac{5}{4}\) by itself: \(\left(\frac{5}{4}\right)\times\left(\frac{5}{4}\right)=\frac{25}{16}\).
04

Simplify the Main Expression

Replace \(\left(\frac{4}{5}\right)^{-2}\) with \(\frac{25}{16}\) in the original expression. So, \(\frac{1}{(\frac{25}{16})}\) can be simplified by taking the reciprocal of \(\frac{25}{16}\), resulting \(\frac{16}{25}\).

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Key Concepts

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

Negative Exponents
Negative exponents can be confusing, but they're just another way to express division. A negative exponent tells you how many times to divide, rather than multiply, a number by itself. For example, a negative exponent like \(a^{-n}\) means you take 1 divided by \(a^n\). In simpler terms, \(a^{-n} = \frac{1}{a^n}\).

Let's apply this to fractions. If we have \((\frac{4}{5})^{-2}\), this negative exponent signals a two-step process. First, flip the fraction, and second, raise it to the power of positive 2.
Reciprocal
Understanding reciprocals can make fraction calculations much simpler. The reciprocal of a number is just the 'flipped' version of it.

For a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). When dealing with fractions raised to negative exponents, you first convert them to their reciprocals.

So, for \(\left(\frac{4}{5}\right)^{-2}\), the reciprocal becomes \(\frac{5}{4}\), allowing the exponent to be positive and easier to manage.
Fractions
Fractions represent a part of a whole and are a simple way to express division. A fraction is composed of a numerator and a denominator. The numerator is the top number, and the denominator is the bottom one.

When you simplify expressions with fractions, you often encounter operations like finding a common denominator or converting complex fractions into simpler ones. In this exercise, when simplifying the given fraction, we use the reciprocal, which turns the division into a multiplication.
Power of a Fraction
Raising a fraction to a power means multiplying the fraction by itself as many times as the power indicates. With positive exponents, this simply involves repeated multiplication. So \(\left(\frac{5}{4}\right)^2\) means \(\left(\frac{5}{4}\right) \times \left(\frac{5}{4}\right)\).

This results in \(\frac{25}{16}\), as you multiply the numerators and the denominators separately: \(5 \times 5 = 25\) and \(4 \times 4 = 16\).

Understanding the process of raising a fraction to a power can make complex expressions easier to handle.

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Most popular questions from this chapter

Convert each number to scientific notation and perform the indicated operations. Express the result in ordinary decimal notation. $$ (0.00003)(0.00025) $$

Perform the indicated operations and express answers in simplest radical form. $$ \frac{\sqrt[3]{16}}{\sqrt[6]{4}} $$

In the year 2008 the public debt of the United States was approximately $$\$ 10,600,000,000,000$$. For July 2008 , the census reported that \(303,000,000\) people lived in the United States. Convert these figures to scientific notation, and compute the average debt per person. Express the result in scientific notation.

Use your calculator to estimate each of the following. Express final answers in ordinary notation rounded to the nearest one-thousandth. (a) \(1.09^{5}\) (b) \(1.08^{10}\) (c) \(1.14^{7}\) (d) \(1.12^{20}\) (e) \(0.785^{4}\) (f) \(0.492^{5}\)

Simplify each of the following. Express final results using positive exponents only. $$ \left(9 x^{2} y^{4}\right)^{\frac{1}{2}} $$
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