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Chapter 4: Problem 28

Evaluate each expression. $$ (-4)^{-3} $$

Short Answer

Expert verified
-\(\frac{1}{64}\)

Step by step solution

01

- Understand Negative Exponents

A negative exponent indicates a reciprocal. For any non-zero number \(a\) and positive integer \(n\), \(a^{-n} = \frac{1}{a^n}\).
02

- Apply the Negative Exponent Rule

Given the expression \((-4)^{-3}\), rewrite it using the negative exponent rule: \((-4)^{-3} = \frac{1}{(-4)^3}\).
03

- Calculate the Exponentiation

Calculate \((-4)^3\). This means multiplying \(-4\) by itself three times: \(-4 \times -4 \times -4\).
04

- Perform the Multiplication

First, multiply \(-4\) by \(-4\), which gives 16. Then multiply 16 by \(-4\), resulting in \(-64\). Thus, \((-4)^3 = -64\).
05

- Substitute Back

Substitute \((-4)^3\) back into the expression: \(\frac{1}{-64}\).
06

- Simplify

The expression \(\frac{1}{-64}\) simplifies to \(-\frac{1}{64}\).

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Key Concepts

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

Reciprocal
A crucial concept in understanding negative exponents is the reciprocal. In mathematics, the reciprocal of a number is 1 divided by that number. For instance, the reciprocal of 4 is \(\frac{1}{4}\). If you have \(\frac{1}{4}\), the reciprocal is 4. This concept is vital when dealing with negative exponents.
When you see a negative exponent, like \(-3\), it simply means to take the reciprocal of the base raised to the positive of that exponent. In the case of \((-4)^{-3}\), we rewrite it as \(\frac{1}{(-4)^3}\).
Remember **:
  • The negative exponent tells you to find the reciprocal.
  • Apply the exponentiation to the new form.
Exponentiation
Exponentiation is the process of raising a number to a power. It involves multiplying the base by itself as many times as indicated by the exponent. In the given exercise, we have \((-4)^3\). This means we need to multiply \-4 \times -4 \times -4\).
Let's break it down:
  • First, multiply \(-4 \times -4 = 16\).
  • Next, take that result and multiply by \(-4) again: \(16 \times -4 = -64\).
Therefore, \((-4)^3 = -64\). This result is crucial for the next step.
By understanding exponentiation, you ensure that you have the correct values to use in further calculations.
Simplifying Fractions
After converting the negative exponent to a reciprocal and performing the exponentiation, we often need to simplify the resulting fraction. For instance, after calculating \((-4)^3 = -64\), we get \(\frac{1}{-64}\).
Let's look at simplification steps:
  • Identify if the fraction can be reduced: In our example, \(\frac{1}{-64}\) is already in its simplest form.
  • Recognize any negative signs: For simplification, this means placing the negative sign correctly. In our case, \(\frac{1}{-64}\) simplifies to \- \frac{1}{64}\.
It is important to clearly express the fraction in its simplest, most understandable form. Processing this step-by-step ensures that you adequately grasp simplifying fractions.

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Most popular questions from this chapter

Find each product. In Exercises \(81-84,89,\) and \(90,\) apply the meaning of exponents. $$ (5 k+3 q)^{2} $$

Find each product. $$ \left(\frac{3}{4}-x\right)\left(\frac{3}{4}+x\right) $$

Explain what it means for a number to be written in scientific notation. Give examples.

In \(2006,\) national expenditures for health care reached 2,106,000,000,000 dollars. Using 300 million as the population of the United States, about how much, to the nearest dollar, was spent on health care person in \(2006 ?\) (Source: U.S. Centers for Medicare and Medicaid Services.)

Our system of numeration is called a decimal system. In a whole number such as 2846 each digit is understood to represent the number of powers of 10 for its place value. The 2 represents two thousands \(\left(2 \times 10^{3}\right),\) the 8 represents eight hundreds \(\left(8 \times 10^{2}\right),\) the 4 represents four tens \(\left(4 \times 10^{1}\right),\) and the 6 represents six ones (or units) \(\left(6 \times 10^{\circ}\right)\) \(2846=\left(2 \times 10^{3}\right)+\left(8 \times 10^{2}\right)+\left(4 \times 10^{1}\right)+\left(6 \times 10^{0}\right) \quad\) Expanded form $$ \text {Keeping this information in mind,} $$ Divide 2846 by \(2,\) using paper-and-pencil methods: \(2 \longdiv { 2 8 4 6 }\)
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