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Chapter 5: Problem 8

Evaluate each expression. See Example 1. $$ (-4)^{3} $$

Short Answer

Expert verified
The expression evaluates to -64.

Step by step solution

01

Understand the Expression

The expression given is \(-4)^{3}\). This represents the number -4 raised to the power of 3. It means multiplying -4 by itself three times.
02

Multiply Base with Itself

The base is -4. To evaluate \(-4)^{3}\), calculate \(-4 imes -4\). This results in \(16\) because multiplying two negative numbers results in a positive number.
03

Continue Multiplying

Now, use the result from Step 2. Multiply the result \(16\) by the base, \(-4\): \(16 imes -4 = -64\). The result is negative because multiplying a positive number by a negative number results in a negative number.
04

Conclude the Calculation

The final result of multiplying \(-4 imes -4\) and then \(16 imes -4\) gives us the evaluated expression value of \(-64\).

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Key Concepts

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

Negative Numbers
Negative numbers might seem confusing at first, but they are essential in mathematics. A negative number is any number less than zero. They are often used to represent loss, debt, or temperature below zero. In an expression like
  • \(-4\)
\(-4\), the negative sign means we are 4 units below zero on a number line.
Negative numbers impact operations like multiplication. If you multiply two negative numbers, the result is positive. However, if you multiply a negative number by a positive number, the result is negative. This rule is critical when dealing with powers of negative numbers.
Multiplication
Multiplication is one of the four basic arithmetic operations. It's like adding a number to itself a certain number of times. For example, multiplying 3 by 4, or
  • \(3 \times 4\)
is the same as adding 3 four times:
\(3 + 3 + 3 + 3 = 12\).
In the expression
  • \((-4)^3\)
you multiply -4 by itself three times. Following the steps is crucial:
  • First, multiply the first two numbers: \(-4 \times -4 = 16\).
  • Then, take this result and multiply it by -4 again: \(16 \times -4 = -64\).
The result negative when multiplying a positive with a negative tells us about the direction or nature of the final output.
Power of Three
Understanding powers, or exponents, is vital in math, especially with powers like three. The power tells you how many times to multiply a number by itself:
  • For \((-4)^3\), the power of three indicates that -4 should be multiplied by itself a total of three times.
Let's break this down easily:
  • The base number \(-4\) is repeated as a factor in the multiplication process,\(-4 \times -4 \times -4\).
The final result from such calculations can be positive or negative, depending on the base and the number of times it is multiplied.
Here, because the base is a negative number and it is multiplied three times (an odd count), the result is negative: \(-64\).
Remember that odd powers of negative numbers always result in negative answers.

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Most popular questions from this chapter

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