Problem 39 - Prealgebra

Question

The problems below review the material on exponents we have covered previously. Expand and simplify. $$5^{3}$$

Step-by-Step Solution

Answer

125

Step 1: Understanding Exponents

The notation \(5^3\) means we use the number 5 as a factor three times. This is because the exponent 3 indicates that 5 is to be multiplied by itself two additional times.

Step 2: Multiplying the Base

To calculate \(5^3\), multiply 5 by itself two times: \(5 \times 5 \times 5\).

Step 3: Calculate the Product

First, multiply the first two 5s: \(5 \times 5 = 25\). Then, take that result and multiply it by the next 5: \(25 \times 5 = 125\). This gives us the expanded and simplified value of \(5^3\).

Key Concepts

Base

In the context of exponents, the "base" is the number that is being multiplied. Understanding the base is fundamental when working with exponential expressions. It is the number on which the operation of repeated multiplication is performed. Consider it as the "foundation" of the expression. When you encounter an exponential expression like \(5^3\), the number 5 is the base.

• The base is the number you focus on spreading across the repeated multiplication.
• In \(5^3\), the base (5) is used three times as shown in the expression \(5 \times 5 \times 5\).
• It's crucial to distinguish the base from the exponent because they determine the value of the expression differently.

In any exponential expression \(b^n\), base \(b\) has to be multiplied by itself \(n-1\) times.

Exponentiation

Exponentiation is the mathematical operation that involves repeated multiplication of a base. It's a concise way to express complex multiplications. The exponent tells you how many times to multiply the base by itself. Understanding the concept of exponentiation is similar to understanding a shortcut in mathematics:

• The notation \(5^3\) is shorthand for saying "multiply 5 by itself 2 more times."
• The exponent (3 in this case) is a small number placed at the upper right of the base number (5).
• This dramatically simplifies the way we express repeated multiplication.

Using exponents, long multiplications become more manageable, especially with larger bases or higher powers.

Multiplication

Multiplication is the arithmetic operation that forms the foundation of exponentiation. In exponentiation, once the base and exponent are identified, you multiply the base by itself as many times as the exponent dictates. This step-by-step multiplication process is crucial for solving exponential expressions:

• Start with the base value, then multiply it with itself based on the number of times the exponent indicates.
• For \(5^3\), calculate it as \(5 \times 5 = 25\), then \(25 \times 5 = 125\).
• The final result from the multiplication is the simplified answer to the exponentiation problem.

By breaking down the multiplication this way, the intimidating problem of exponentiation becomes manageable and straightforward.