Problem 27 - Fundamentals of Mathematics

Question

Expand the terms. (Do not find the actual value.) \(117^{5}\)

Step-by-Step Solution

Answer

\(117 \times 117 \times 117 \times 117 \times 117\).

Step 1: Set the Base Expression

We are given the expression \(117^5\), and our task is to expand it without finding the actual numerical value. So, note that our base is 117 and our exponent is 5.

Step 2: Understand Expansion with Exponents

The power of 5 means multiplying the base (117) with itself 5 times. This forms the fundamental idea of expanding a power expression.

Step 3: Write Out the Full Expansion

Express \(117^5\) as a multiplication of five terms: \(117 \times 117 \times 117 \times 117 \times 117\).

Step 4: Note About Expansion

In expansion problems like this, our goal is to retain the expression in its expanded form for clarity, rather than compute to get a single number.

Key Concepts

Power Expression

A power expression is a compact way of expressing repeated multiplication. It has two main parts: a base and an exponent. In expressions like \(117^5\), it helps represent large computations in a simplified manner. The power expression \(117^5\) is not just a number; it conveys the operation of multiplying 117 by itself a total of five times. This is particularly useful when dealing with very large numbers, as it helps in visualizing the process without getting lost in calculations. By expanding power expressions, you reveal the entire process behind the operation, which becomes important for better understanding and application in algebra and other areas of mathematics.

Base and Exponent

The base and the exponent are the two elements that make up a power expression.

• Base: This is the number that will be multiplied. In \(117^5\), 117 is the base.
• Exponent: This indicates how many times the base is used as a factor. In the expression \(117^5\), the number 5 is the exponent.

The significance of these two components cannot be overstated. The base shows what is being multiplied, while the exponent shows how many times the base appears in the multiplication sequence. Together, they create a powerful tool for handling large operations efficiently. By understanding each of these parts, you can better navigate more complex expressions.

Multiplication of Terms

Multiplication of terms in an expanded power expression involves writing the base the number of times indicated by the exponent and placing a multiplication sign between each instance. For example, \(117^5\) is expanded to \(117 \times 117 \times 117 \times 117 \times 117\). This visualization is crucial:

• Step-by-Step Process: Each multiplication step is a part of expressing repeated multiplication from the power expression. While you don't find the product, you clearly see the multiple instances of the base.
• Understanding Growth: Realizing that each added exponent step multiplies the outcome by the base itself offers insight into comprehending exponential growth patterns.

Writing out the multiplication of terms showcases the number of times the base is used, reinforcing the concept of exponentiation and allowing you to further explore more abstract mathematical concepts without needing to initially calculate the full numerical result.