Problem 1 - Elementary and Intermediate Algebra

Question

In the exponential expression \(7^{5}, 7\) is the_____ \(\quad,\) and 5 is the____ \(.7^{5}\) is the fifth_____ \(\quad\) of seven.

Step-by-Step Solution

Answer

In \(7^{5}\), 7 is the base, 5 is the exponent, and it's the fifth power of 7.

Step 1: Identify the Base

In the expression \(7^{5}\), 7 is the base. The base is the number that is repeatedly multiplied according to the exponent.

Step 2: Identify the Exponent

In the expression \(7^{5}\), the number 5 is the exponent. The exponent indicates how many times the base is multiplied by itself.

Step 3: Identify the Power

The expression \(7^{5}\) is referred to as the fifth power of 7. This indicates that 7 is multiplied by itself 5 times.

Key Concepts

Base

In an exponential expression like \(7^5\), the base is the number that is being repeatedly multiplied. Here, the number 7 is called the base. It's crucial to understand that the base is the foundation of the exponentiation process.
When you see something like \(b^n\), the base \(b\) is the number you start with, and it’s the number that will be multiplied by itself, again and again, as many times as indicated by the exponent. This is why we call it the base—it’s the basis for all the multiplication in the expression.
Here's a quick summary:

• The base is the number that sits at the "bottom" of the exponent expression.
• It is the number being multiplied by itself over and over.
• In \(7^5\), the base is 7.

Exponent

The exponent in an exponential expression is the small number written above and to the right of the base. In our example, \(7^5\), the number 5 is the exponent. Essentially, an exponent tells us how many times the base is used as a factor.
Think of it as a shorthand or a quick way to represent repeated multiplication. Instead of writing \(7 \times 7 \times 7 \times 7 \times 7\), we use the compact form \(7^5\).
Key points about exponents:

• The exponent shows how many times the base is multiplied by itself.
• It provides a concise way to express larger operations or processes.
• In \(7^5\), 5 is the exponent, which instructs us to multiply 7 by itself 5 times.

Power

The term "power" in mathematics refers to the entire expression involving a base and an exponent. For example, in \(7^5\), the expression is known as "the fifth power of 7."
Here, "fifth" refers to the power and it speaks to the multitude of the multiplication process. The power tells us not just about multiplication, but about a sequence of operations. In specific terms, the power shows the final result or expression of multiplying the base repeatedly as dictated by the exponent.
To break it down further:

• A power combines the base and exponent into a single mathematical concept.
• It indicates the result of repeated multiplication of the base.
• \(7^5\) is called the fifth power of 7, meaning 7 is used in multiplication 5 times in sequence.

Understanding how the base, exponent, and power work together will simplify complex problems and enable efficient calculation.