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Chapter 2: Problem 333

In the following exercises, write in expanded form. $$ 8^{4} $$

Short Answer

Expert verified
8 \times 8 \times 8 \times 8

Step by step solution

01

Understand the Exponential Notation

The expression we have to expand is \(8^4\). This is an exponential notation where 8 is the base and 4 is the exponent, meaning that 8 is multiplied by itself 4 times.
02

Expand the Exponential Expression

To express \(8^4\) in expanded form, write it as the product of 8 multiplied by itself 4 times: \[8^4 = 8 \times 8 \times 8 \times 8\].

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Key Concepts

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

Expanded Form
When dealing with exponential notation, converting it into expanded form can help you better understand the concept. Expanded form is simply writing out the multiplication process suggested by an exponent.

For instance, consider the expression given in the exercise: \(8^4\). The formula tells us to multiply the base, which is 8, by itself as many times as specified by the exponent, which is 4 times.

So, in expanded form, \(8^4\) becomes: \[8^4 = 8 \times 8 \times 8 \times 8.\]

Breaking down the multiplication in this way helps you to visualize each step involved.
Base and Exponent
Understanding the components of an exponential expression is crucial for properly expanding it. In our example, \(8^4\), the number 8 is called the 'base' and the number 4 is the 'exponent.'
  • The base (8) is the number being multiplied.
  • The exponent (4) indicates how many times the base is used as a factor in the multiplication.
Let's look at another example to make it clearer:

For \(2^3\), the base is 2 and the exponent is 3.
In expanded form, it is \[2^3 = 2 \times 2 \times 2\]

So, the exponent tells you the number of times to multiply the base by itself. This concept is straightforward once you know what each term represents.
Multiplication
Multiplication is a fundamental arithmetic operation that links closely with the concept of exponents. When performing multiplication for exponential expressions, you are repeatedly multiplying the base by itself.

In our original problem, we have \[8^4 = 8 \times 8 \times 8 \times 8.\] This means:
  • First, multiply the first two 8s: \[8 \times 8 = 64.\]
  • Next, multiply the result by the next 8: \[64 \times 8 = 512.\]
  • Finally, multiply the new result by the last 8: \[512 \times 8 = 4096.\]
Thus, \[8^4 = 4096.\]

This process shows the power of exponents and how they simplify repeated multiplication into a more compact form.

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