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Chapter 1: Problem 7

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

Short Answer

Expert verified
-64

Step by step solution

01

Identify the base and the exponent

In the expression \((-4)^{3}\), the base is -4 and the exponent is 3.
02

Apply the exponentiation

Raise the base to the power of the exponent. Since it's raised to an odd power, the result will be negative. Specifically, \((-4)^{3} = -4 * -4 * -4 = -64.\)

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Key Concepts

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

Understanding Base and Exponent
Exponential expressions have two main parts: the base and the exponent. The base is the number that is being multiplied by itself. The exponent tells us how many times to multiply this base. In the expression \((-4)^3\), the base is \(-4\), and the exponent is \(3\). This means we multiply \(-4\) by itself three times: \(-4 \times -4 \times -4\). Each multiplication uses the base as a factor.
  • Base: The number being multiplied repeatedly. In this case, \(-4\).
  • Exponent: Indicates how many times the base is used as a factor. For \(3\), it's used three times.
Recognizing these components is key to understanding and evaluating exponential expressions. The base and exponent work together to shape the value that the expression ultimately represents.
Negative Numbers in Exponents
When a negative number is used as the base in an exponential expression, it behaves differently than a positive base. The combination of negative and exponent impacts the evaluation of the expression. When the base is negative and the exponent is a positive integer, the sign of the result depends on whether the exponent is odd or even.
  • Negative Base: Shows that the same negative number is being multiplied repeatedly.
  • Sign: Determined by the odd or even nature of the exponent.
For \((-4)^3\), which involves an odd exponent, the process keeps the result negative due to an odd number of negative multiplications. Understanding how negative numbers work with exponents helps prevent mistakes and ensures accurate calculations.
Odd and Even Exponents
The exponent鈥檚 value as odd or even significantly alters an exponential expression's result. With negative bases, its effect is particularly noticeable.
  • Odd Exponents: When an exponent is odd, like \(3\) in \((-4)^3\), the result of the expression is negative. Multiplying an odd number of negative factors results in a negative product: \[-4 \times -4 \times -4 = -64\].
  • Even Exponents: An even exponent, however, would produce a positive outcome. For instance, \((-4)^2 = -4 \times -4 = 16\).
Being able to identify whether an exponent is odd or even will allow you to predict the sign of the result for given bases. This understanding provides greater control over manipulating and solving complex exponential expressions.

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Most popular questions from this chapter

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