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Chapter 1: Problem 3

In Exercises \(1-14\), evaluate each exponential expression. $$4^{3}$$

Short Answer

Expert verified
The evaluation of the expression \(4^{3}\) is 64.

Step by step solution

01

Understand the Exponential Expression

When looking at \(4^{3}\), it is understood to mean 'four to the power of three.' This means 4 is the base and 3 is the exponent.
02

Apply the Rules of Exponents

An exponent refers to the number of times a number is multiplied by itself. So, \(4^{3}\) means 4 is multiplied by itself three times. That is, \(4 * 4 * 4\).
03

Perform the Calculation

Now, multiply the base (4) by itself three times. So, \(4 * 4 * 4 = 64\).

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Key Concepts

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

Exponents
Exponents are a fundamental concept in mathematics that indicate how many times a number is to be multiplied by itself. When you see an exponential expression such as \(4^3\), the exponent (in this case 3) tells you that the base number (4) should be multiplied by itself a total of three times.

Imagine exponents as a form of shorthand that simplifies the expression of large numbers or repeated multiplications. Instead of writing \(4 \times 4 \times 4\), you write \(4^3\). This is both a time-saver and a way to avoid confusion in calculations. Exponents are not limited to integers, and learning how to manipulate them is crucial for algebra and higher-level mathematics.
Base and Exponent
In any exponential expression, such as \(4^3\), you'll find two main components: the base and the exponent. Understanding these components is essential to evaluating exponential expressions correctly.

The base, represented here as 4, is the number that will be repeatedly multiplied. It's the "foundation" in our expression. Meanwhile, the exponent, shown as 3, indicates how many times you multiply the base by itself. By following the instruction given by the exponent, you apply the operation of multiplication in a specific manner.
  • The Base: The main number being multiplied.
  • The Exponent: The power to which the base is raised.
Learning to identify and correctly use the base and exponent is key to mastering exponential expressions.
Multiplication
Multiplication in the context of exponents involves repeated operations on the base number as specified by the exponent. When dealing with an expression like \(4^3\), you are performing multiplication by repeating the base number, 4, until you have achieved the required number of operations.

So, in \(4^3\), you multiply 4 by 4 to get 16, and then again multiply by 4 to reach 64. Essentially, you have:
  • 4 first time: 4
  • 4 second time: 4 \(\times\) 4 = 16
  • 4 third time: 16 \(\times\) 4 = 64
This step-by-step approach helps in understanding how each multiplication relates directly to the exponent's instruction, reinforcing both your multiplication skills and comprehension of exponential expressions.

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