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Chapter 5: Problem 57

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

Short Answer

Expert verified
The value of \(4^{3}\) is 64.

Step by step solution

01

Understand the Exponential Expression

In the expression \(4^{3}\), 4 is the base and 3 is the exponent. The exponent tells us how many times to multiply the base by itself. So, \(4^{3}\) means \('4 multiplied by itself 3 times'\).
02

Perform the Multiplication

Multiply the base by itself as many times as the exponent states. This operation will look like this: \(4*4*4\).
03

Calculation & Final Answer

After performing the multiplication \(4*4*4\), the total equals 64. Therefore, \(4^{3} = 64\).

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Key Concepts

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

Exponential Expressions
When dealing with exponential expressions, it is all about understanding how repeated multiplication works. An exponential expression consists of two main parts: a base and an exponent. The base is the number that we multiply by itself, and the exponent dictates how many times this multiplication occurs.
In expressions like these, such as \(4^3\), the goal is to simplify or evaluate the expression. The expression indicates that the base, 4, should be multiplied by itself according to the exponent, which is 3 times.
This is a powerful concept in mathematics because it allows us to express large numbers more compactly and effectively.
Base and Exponent
In mathematics, distinguishing between the base and the exponent in an expression is crucial for accurately performing operations. In the expression \(4^3\), the number 4 is referred to as the base, and the number 3 is the exponent.
Understanding the roles of the base and exponent helps us comprehend how the exponential expression will be evaluated:
  • The base tells us which number we are repeatedly multiplying.
  • The exponent tells us how many times the base is used as a factor in the multiplication process.

So, when we see \(4^3\), we translate this as multiplying 4 three times because of the exponent 3.
Multiplication
Multiplication within exponential expressions involves using the base as a factor multiple times. This is the core process of evaluating expressions like \(4^3\).
To carry out the multiplication in the scenario of \(4^3\), we write it down as \(4 \times 4 \times 4\). Following this:
  • The first step is to multiply 4 by 4, which gives us 16.
  • Next, we take this result and multiply it again by 4, arriving at 64.

This systematic approach ensures that the expression is fully simplified and calculated correctly. Performing the multiplication this way yields the final solution of 64, illustrating how exponential expressions compound values efficiently.

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Most popular questions from this chapter

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(\frac{2}{3}, 1, \frac{4}{3}, \frac{5}{3}, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=\frac{1}{4}, r=\frac{1}{2}\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{6}\), when \(a_{1}=-2, r=-3\).

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-\frac{1}{8}, r=-2\)

A person is investigating two employment opportunities. They both have a beginning salary of $$\$ 20,000$$ per year. Company A offers an increase of $$\$ 1000$$ per year. Company B offers \(5 \%\) more than during the preceding year. Which company will pay more in the sixth year?
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