Problem 20 - Algebra for College Students

Question

For Problems 13-50, perform the indicated operations involving rational expressions. Express final answers in simplest form. \(\frac{7 x^{2} y}{9 x y^{3}} \div \frac{3 x^{4}}{2 x^{2} y^{2}}\)

Step-by-Step Solution

Answer

The simplest form is \( \frac{14}{27x} \)."

Step 1: Rewrite the Division as Multiplication

The expression \( \frac{7 x^2 y}{9 x y^3} \div \frac{3 x^4}{2 x^2 y^2} \) can be rewritten by converting the division into multiplication by the reciprocal. It becomes \( \frac{7 x^2 y}{9 x y^3} \times \frac{2 x^2 y^2}{3 x^4} \).

Step 2: Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \( (7 x^2 y) \cdot (2 x^2 y^2) = 14 x^4 y^3 \) and \( (9 x y^3) \cdot (3 x^4) = 27 x^5 y^3 \). The expression now is \( \frac{14 x^4 y^3}{27 x^5 y^3} \).

Step 3: Simplify the Expression by Canceling Common Factors

The terms \( y^3 \) in both the numerator and the denominator can be canceled out. This simplifies the expression to \( \frac{14 x^4}{27 x^5} \).

Step 4: Simplify Further by Reducing Common Variables

Now reduce \( x^4 \) with \( x^5 \) in the denominator: \( \frac{x^4}{x^5} = \frac{1}{x^{5-4}} = \frac{1}{x} \). Hence, the expression simplifies to \( \frac{14}{27x} \).

Key Concepts

Simplifying Rational Expressions

Simplifying rational expressions involves reducing them to their simplest form by removing common factors in the numerator and the denominator. Think of it like reducing fractions. For example, if you have the expression \( \frac{10}{15} \), you can simplify it by dividing both the numerator and denominator by their greatest common factor, which is 5, to get \( \frac{2}{3} \). Similarly, to simplify a rational expression:

• Factor both the numerator and denominator completely.
• Cancel out any common factors they share.

Let's consider a rational expression \( \frac{14 x^4}{27 x^5} \). The common factor here is \( x^4 \), so you can cancel it, simplifying the expression to \( \frac{14}{27x} \). This tells us that we only need to look for values not equal to zero for \( x \) to retain the expression's validity.

Multiplying Rational Expressions

Multiplying rational expressions is a bit like multiplying regular fractions. When you multiply two fractions, you multiply the numerators together and the denominators together. For example, \( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \). When multiplying rational expressions:

• Multiply the numerators together.
• Multiply the denominators together.

For example, consider multiplying the expressions \( \frac{7 x^2 y}{9 x y^3} \) and \( \frac{2 x^2 y^2}{3 x^4} \). Multiply the numerators: \( 7 x^2 y \times 2 x^2 y^2 = 14 x^4 y^3 \). Then, multiply the denominators: \( 9 x y^3 \times 3 x^4 = 27 x^5 y^3 \). The resulting expression is \( \frac{14 x^4 y^3}{27 x^5 y^3} \). Simplifying can then make it easier to work with later.

Division of Rational Expressions

Dividing rational expressions is very similar to dividing whole numbers—by multiplying by the reciprocal. When dividing rational expressions:

• Find the reciprocal of the divisor.
• Multiply the dividend by this reciprocal.

Suppose you have the division problem \( \frac{7 x^2 y}{9 x y^3} \div \frac{3 x^4}{2 x^2 y^2} \). You convert it into multiplication by taking the reciprocal of \( \frac{3 x^4}{2 x^2 y^2} \), which is \( \frac{2 x^2 y^2}{3 x^4} \). Thus, the division becomes \( \frac{7 x^2 y}{9 x y^3} \times \frac{2 x^2 y^2}{3 x^4} \). Now you can apply multiplication steps to complete your work and then simplify the final expression.