Problem 12 - Elementary and Intermediate Algebra

Question

Simplify. $$ 5^{3} $$

Step-by-Step Solution

Answer

125

Step 1 - Understand the exponent

The expression given is using an exponent: 5^3. Here, 3 is the exponent which indicates how many times to multiply the base number 5 by itself.

Step 2 - Multiply the base number

Perform the multiplication: 5^3 = 5 * 5 * 5.

Step 3 - Calculate the product

Now calculate the product of these multiplications: First, 5 * 5 = 25. Then, 25 * 5 = 125.

Step 4 - Write the simplified result

Therefore, the simplified result of 5^3 is 125.

Key Concepts

Understanding the Base Number

In exponentiation, a 'base number' is the number that we are multiplying. It is the number we raise to a power. In the expression provided, 5 is the base number.

To visualize it better:

• In the exponentiation expression 5^3, 5 is the number we start with.
• We will multiply 5 by itself a number of times indicated by the exponent.

So, when you see 5^3, think of 5 as the fundamental building block that's being repeatedly used in multiplication to get a larger number. This foundation number is crucial for understanding how the entire calculation will play out.

The Role of the Exponent

An 'exponent' tells us how many times to use the base number in a multiplication.

In the expression 5^3, the exponent is 3. This indicates that the base number 5 should be multiplied by itself three times.

To break it down:

• The expression 5^3 signifies 5 * 5 * 5.
• The exponent helps us know how many times to multiply the base number.

Essentially, the exponent provides the details about how much repetition of multiplication is needed to compute the final product.

Steps of Multiplication in Exponentiation

Multiplication is at the core of exponentiation. Here’s how it unfolds step-by-step:

Let's use the example from our exercise, 5^3.

• First, think of it as multiplying the base number 5 by itself. The exponent 3 shows us how many times we need to do the multiplication.
• Start by multiplying two of them: 5 * 5 = 25.
• Next, use the product (25) and multiply by the base number again: 25 * 5 = 125.

This method of breaking down the multiplication stepwise makes the calculation straightforward and simpler to approach.

So by following the exact steps:

• 5 * 5 = 25
• 25 * 5 = 125

we reach the result of 5^3 being 125.