Problem 17 - Algebra for College Students

Question

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{5 x y}{8 y^{2}} \cdot \frac{18 x^{2} y}{15}\)

Step-by-Step Solution

Answer

\(\frac{3x^3}{4}\)

Step 1: Rewrite the Expression

The given expression is \( \frac{5xy}{8y^{2}} \cdot \frac{18x^{2}y}{15} \). This can be rewritten as a single fraction:\[\frac{5xy \cdot 18x^{2}y}{8y^{2} \cdot 15}.\]

Step 2: Multiply the Numerators

Multiply the numerators together:\[5xy \times 18x^{2}y = 90x^{3}y^{2}.\] This gives us the new numerator.

Step 3: Multiply the Denominators

Multiply the denominators together:\[8y^{2} \times 15 = 120y^{2}.\] This gives us the new denominator.

Step 4: Combine and Simplify the Expression

Now, place the new numerator and denominator into a single fraction:\[\frac{90x^{3}y^{2}}{120y^{2}}.\]Next, simplify the expression by canceling out common terms. Both the numerator and denominator have \(y^{2}\), which cancels out. Simplify the constants by dividing both 90 and 120 by their GCD, which is 30.\[\frac{90}{120} = \frac{3}{4}.\] Thus, the simplified form is:\[\frac{3x^{3}}{4}.\]

Step 5: Validate the Simplification

Verify that the fraction is simplified by ensuring there are no common factors between the numerator and the denominator. The numerator is \(3x^{3}\) and the denominator is 4; there are no common factors. Therefore, \(\frac{3x^{3}}{4}\) is indeed in simplest form.

Key Concepts

Multiplying Rational Expressions

When working with rational expressions, one of the fundamental operations is multiplication. A rational expression is essentially a fraction that involves polynomials. To multiply rational expressions like \( \frac{5xy}{8y^2} \cdot \frac{18x^2y}{15} \), you need to multiply the numerators together and the denominators together separately, similar to how you would multiply regular fractions.

**Steps to Multiply:**

• Multiply the numerators: Combine all terms from each numerator to form a new numerator, as seen with \( 5xy \times 18x^2y = 90x^3y^2 \).
• Multiply the denominators: Do the same for denominators, like \( 8y^2 \times 15 = 120y^2 \).

After multiplying, you will often need to simplify the resulting fraction to ensure it is in its simplest form.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing the expression to its most concise form so that it doesn't contain any common factors other than 1. After multiplying, as in our problem, you have the expression \( \frac{90x^3y^2}{120y^2} \).

**Steps to Simplify:**

• Identify common terms or factors in the numerator and denominator, like \( y^2 \).
• Cancel out the common terms across the numerator and the denominator: Both have \( y^2 \), so canceling simplifies to \( \frac{90x^3}{120} \).
• Find the greatest common divisor (GCD) of the remaining terms' coefficients and divide: \( \frac{90}{120} \) simplifies to \( \frac{3}{4} \), because their GCD is 30.
• Write the simplified expression: This results in \( \frac{3x^3}{4} \).

Always check your final expression to make sure there are no further common factors between the numerator and denominator.

Algebraic Fractions

Algebraic fractions are similar to numerical fractions, but they include variables as part of their numerators, denominators, or both. Rational expressions, or algebraic fractions, follow the same fundamental principles of arithmetic as regular fractions.

**Key Points:**

• Variables are treated as factors; the same cancellation of common terms applies.
• The laws of exponents are utilized when handling expressions with powers, such as adding or multiplying exponents when terms are multiplied together.
• Ensure that no variable in the denominator is zero, as division by zero is undefined. Always consider the domain of the expression.

Understanding algebraic fractions is essential for manipulating and solving problems involving rational expressions effectively.