Problem 33 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{1}{x^{2}} \cdot \frac{3 x}{2} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3}{2x} \).

Step 1: Write the Expression

The given expression is \( \frac{1}{x^{2}} \cdot \frac{3x}{2} \). We need to multiply these two fractions.

Step 2: Multiply the Numerators

First, multiply the numerators of the fractions. The numerators are 1 and \( 3x \). So, the product of the numerators is \( 1 \cdot 3x = 3x \).

Step 3: Multiply the Denominators

Next, multiply the denominators. The denominators are \( x^{2} \) and 2. So, the product of the denominators is \( x^{2} \cdot 2 = 2x^{2} \).

Step 4: Combine the Resulting Fraction

Combine the results from Steps 2 and 3 into one fraction: \( \frac{3x}{2x^{2}} \).

Step 5: Simplify the Fraction

To simplify \( \frac{3x}{2x^{2}} \), cancel common factors in the numerator and the denominator. Both have an \( x \), which simplifies to \( \frac{3}{2x} \).

Key Concepts

Fraction Multiplication

Fraction multiplication is a straightforward process that involves multiplying the numerators and denominators of fractions. When you multiply fractions, you take the top numbers (numerators) and multiply them together. Similarly, you do the same for the bottom numbers (denominators).
To multiply two fractions:

• Multiply the numerators to get the new numerator.
• Multiply the denominators to get the new denominator.

For our example, multiplying \( \frac{1}{x^{2}} \) by \( \frac{3x}{2} \) involves:

• Numerators: \( 1 \cdot 3x = 3x \)
• Denominators: \( x^{2} \cdot 2 = 2x^{2} \)

Thus, after combining, we get the fraction \( \frac{3x}{2x^{2}} \). Remember to focus on accuracy during multiplication to avoid errors in subsequent steps.

Simplifying Expressions

Simplifying expressions is about making them less complex while maintaining their value. In rational expressions, as with regular algebraic expressions, simplification often involves canceling common factors in the numerator and denominator. This process can make expressions easier to interpret and solve.
To simplify an expression like \( \frac{3x}{2x^{2}} \):

• Identify common factors in the numerator and denominator. Here, both have a factor of \( x \).
• Cancel the common factor across the fraction.

This leads to simplification: \( x \) in the numerator and one of the \( x's \) in \( x^{2} \) in the denominator can be canceled. After canceling, the expression becomes \( \frac{3}{2x} \). Simplification helps provide a clearer understanding of the expression.

Rational Expressions

Rational expressions involve fractions made up of polynomials in both the numerator and the denominator. These expressions follow the same rules as ordinary fractions but may require extra steps to simplify.
A rational expression generally looks like \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. The process of working with rational expressions includes:

• Performing arithmetic operations, such as addition, subtraction, multiplication, and division.
• Simplifying by factorization or canceling common factors.

In our exercise, \( \frac{3x}{2x^{2}} \) is already a rational expression from fractional multiplication. Simplifying it as shown in earlier steps results in \( \frac{3}{2x} \). Mastery of rational expressions is crucial in algebra as it helps in solving complex equations efficiently.