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Chapter 1: Problem 30

Evaluate. $$-4^{3}$$

Short Answer

Expert verified
The expression \(-4^3\) evaluates to -64 by following these steps: Identify the base (-4) and exponent (3), multiply the base by itself for the number of times given by the exponent (\((-4) \times (-4) \times (-4)\)), and perform the multiplication to get the result, which is -64.

Step by step solution

01

Identify the base and exponent

We're given \(-4^3\), where -4 is the base and 3 is the exponent. We need to multiply the base by itself for the number of times given by the exponent.
02

Calculate the exponentiation

We will multiply the base -4 by itself 3 times: \((-4) \times (-4) \times (-4)\)
03

Evaluate the multiplication

Now, let's multiply the numbers together. First, we multiply -4 by -4: \((-4) \times (-4) = 16\) Next, we multiply the result by -4: \(16 \times (-4) = -64\)
04

Write the final result

The expression \(-4^3\) evaluates to -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
In mathematics, the concept of base and exponent is foundational for exponentiation. The base is the number that we multiply by itself. The exponent tells us how many times to multiply the base by itself.
For example, in the expression \(-4^3\), -4 is the base, and 3 is the exponent. This means that -4 needs to be multiplied by itself three times, noting that it treats the 3 as a command for how many times the base is used in a repeated multiplication:
  • The first multiplication: -4
  • The second multiplication: -4 multiplied by -4
  • The third and final multiplication: result of the previous step multiplied by -4
Understanding base and exponent helps to structure these repeated operations clearly. Remember that the base in our case, even though negative, follows the same multiplicative rule.
Working With Negative Numbers
Negative numbers can sometimes be tricky, especially when involved in exponentiation.
When dealing with negative bases, we need to be mindful of whether the exponent is odd or even. In our example \(-4^3\), the base is negative, and the exponent is 3鈥攁n odd number.
  • If the exponent is odd, the result will also be negative as the negative sign does not cancel out.
  • Conversely, if the exponent were even, the result would be positive because multiplying two negative numbers results in a positive product.
This principle is helpful in predicting the sign of the final product before performing the actual multiplication. Thus, with a base of -4 and an odd exponent, the outcome remains negative.
Applying Order of Operations
Order of operations is a vital arithmetic rule that guides the sequence in which operations should be performed.
Commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), it ensures expressions are solved correctly.
  • In the expression \(-4^3\), addressing the exponentiation occurs first before considering any other arithmetic operation.
  • Notice that the exponentiation involves only the number 4 in this context; thus, the negative sign is not part of the exponentiated value when written as \(-4^3\). As a result, the actual calculation is similar to applying \(-(4^3)\).
This differentiation underlines why \(-4^3\) yields -64: the negative sign isn't part of the repeated multiplication. Understanding order of operations ensures accuracy when dealing with such expressions.

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