Problem 68 - Elementary and Intermediate Algebra

Question

Use the power of a product rule for exponents to simplify each expression. $$ (3 b)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(27b^3\).

Step 1: Understand the Power of a Product Rule

The power of a product rule states that when you raise a product to a power, you can apply the exponent to each factor in the product individually. Mathematically, it is expressed as \((ab)^n = a^n imes b^n\).

Step 2: Apply the Rule to the Expression

We apply the power of a product rule to the expression \((3b)^3\). According to the rule, this becomes \(3^3 \times b^3\).

Step 3: Calculate Individual Exponents

Now calculate the individual exponents: \(3^3 = 27\) because \(3 \times 3 \times 3 = 27\). The expression \(b^3\) stays as is.

Step 4: Combine the Results

Combine the results from step 3: the expression becomes \(27b^3\).

Key Concepts

Power of a Product Rule

When working with exponents, one of the most useful rules is the Power of a Product Rule. This rule helps simplify expressions where a product is raised to a power. By applying this rule, you can simplify an expression like

• \((ab)^n = a^n \times b^n\)

Where instead of multiplying \(a\) and \(b\) first and then raising the result to a power, you can distribute the exponent to each factor individually. Consider the expression

• \((3b)^3\)

Using the Power of a Product Rule, this expression can be rewritten by applying the exponent 3 to both 3 and \(b\), becoming

• \(3^3 \times b^3\)

This distribution makes it easier to handle each part separately. Understanding this rule is critical when simplifying expressions involving exponents.

Simplifying Expressions

Simplifying expressions is a vital skill in algebra that involves reducing the expression into its simplest form. This process helps both in managing complex algebraic problems and in finding equivalent expressions more easily.
To simplify an expression like

• \((3b)^3\)

First, apply the Power of a Product Rule. After applying the rule, you have

• \(3^3 \times b^3\)

Next, you calculate any numerical exponents. For example, calculating \(3^3\) involves multiplying 3 by itself three times:

• \(3 \times 3 \times 3 = 27\)

Leaving the variable part \(b^3\) as it is, the expression simplifies to

• \(27b^3\)

Simplifying expressions involves these straightforward steps: applying rules, performing arithmetic operations, and leaving variables in exponent form.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. These are foundational in algebra and are used to represent mathematical relationships and unknown values.
Examples of expressions include terms like

• \((3b)\)
• \(b^2 + 4b + 4\)
• \((a+b)^2 - c\)

Simplifying algebraic expressions often involves rules like the Power of a Product Rule. For instance, with

• \((3b)^3\)

Knowing this rule helps break down expressions to a simpler form. Expressions may also involve factoring or expanding, which depend on understanding the relationships between the terms. Mastering algebraic expressions is essential for progressing in mathematical studies, as they form the groundwork for understanding equations and functions.