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Chapter 1: Problem 17

Expand the exponential expression as a repeated multiplication. Do not simplify. $$4^{8}$$

Short Answer

Expert verified
\(4^8 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\).

Step by step solution

01

Identify the Base and the Exponent

The given expression is \(4^8\). Here, 4 is the base and 8 is the exponent.
02

Understand the Meaning of the Exponent

An exponent tells us to use the base as a factor in a repeated multiplication. So \(4^8\) means multiplying 4 by itself 8 times.
03

Write as Repeated Multiplication

Since the exponent is 8, write the base (4) multiplied by itself 8 times: \(4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\).

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Key Concepts

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

Base and Exponent
When working with exponential expressions, it's essential to understand their components. An exponential expression comprises a base and an exponent. The base is the number that gets multiplied. The exponent, on the other hand, tells us how many times the base is used as a factor in the multiplication.

For instance, in the expression \(4^8\), 4 is known as the base while 8 is the exponent. Knowing which number is the base and which is the exponent is crucial for correctly interpreting and working with exponential expressions.
Repeated Multiplication
An exponent signifies repeated multiplication of the base. This means you multiply the base by itself as many times as the exponent indicates.

For example, consider the expression \(4^8\). The exponent is 8, so you repeat the base (which is 4) in the multiplication 8 times: \(4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4\). This method is what we call repeated multiplication. It simplifies calculations and helps in easily representing large numbers.
Powers of Numbers
Exponential expressions can also be referred to as powers of numbers. When we say '4 to the power of 8' or '4 raised to the eighth power,' we are discussing the expression \(4^8\).

The terminology 'power of numbers' directly relates to the process of repeated multiplication. By understanding that \(4^8\) represents '4 multiplied by itself eight times,' we grasp how powers help break down and simplify large multiplications.
  • The base (4) is the repeated factor.
  • The exponent (8) tells us that the base is used 8 times in multiplication.
Recognizing powers of numbers is a key skill in working with exponential expressions, making large-scale calculations more feasible and less error-prone.

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