Problem 34 - College Algebra

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. $$ (l \times w)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(l^2 \times w^2\).

Step 1: Identify the Expression

The expression given is \((l \times w)^2\). Our goal is to simplify it.

Step 2: Apply the Power of a Product Rule

According to the power of a product rule, \((a \times b)^n = a^n \times b^n\). In this case, apply it to \((l \times w)^2\), resulting in \(l^2 \times w^2\).

Step 3: Simplify and Ensure Positive Exponents

After applying the rule, we have \(l^2 \times w^2\). This expression already has positive exponents, so no further changes are needed.

Key Concepts

Exponentiation

Exponentiation is a fundamental concept in algebra that involves raising a number or variable to a power. It's a shorthand for repeated multiplication of a number by itself. For example, when we have the base number "a" raised to the power of "n" (written as \( a^n \)), it means multiplying "a" by itself "n" times. Here are the key points to remember about exponentiation:

• The base is the number being multiplied.
• The exponent indicates how many times the base is used as a factor.
• For instance, \( l^2 \) means "l multiplied by l" (or "l times l").

Applying this concept to expressions, we can simplify them or manipulate their parts for solving equations. It's important to be comfortable with exponentiation as it appears frequently in algebraic operations and many areas of mathematics beyond.

Power of a Product Rule

The power of a product rule is a specific rule used in exponentiation which simplifies expressions where a product is raised to a power. This rule is essential when dealing with algebraic expressions containing multiple variables or numbers within the parentheses, each raised to the same exponent. The rule states:\[ (a \times b)^n = a^n \times b^n \]This means you distribute the power to each factor inside the parentheses. Let's break down how this works:

• When you have a product like \((l \times w)^2\), apply the exponent to each element within the parentheses.
• This results in \(l^2 \times w^2\).
• Each variable is handled separately as if raised independently to the power outside.

Understanding and using this rule allows for efficient simplification of complex algebraic expressions and helps streamline computations.

Simplification of Expressions

Simplification is a process used in algebra to rewrite an expression in its simplest or most reduced form. The goal is to make expressions easier to work with without changing their value. Simplification helps in solving equations and understanding mathematical relationships more clearly.

• To simplify an expression like \((l \times w)^2\), use rules like the power of a product to break it into simpler parts.
• After application, the expression becomes \(l^2 \times w^2\) which is often easier to interpret or compute with.
• Ensuring positive exponents is a part of this process, as expressions are generally preferred to be presented this way unless otherwise needed.

Simplifying expressions aids in comparing, solving, and deriving relationships in algebraic contexts. It's an indispensable skill in mathematics, particularly when dealing with large expressions or complex equations.