# 7.4 – Multiplying and Dividing Rational Expressions

## Key Terms

• No key terms for this section.

## Review

Multiplying Fractions
• Multiply straight across the numerator, then straight across the denominator.
• Ex. $\frac{4}{5}\bullet \frac{3}{7}=\frac{4\; \bullet \; 3}{5\; \bullet \; 7}=\frac{12}{35}$
• If a fraction can be reduced, you may reduce before you multiply or after.
• Ex. Before (reduce, then multiply): $\frac{2}{3}\bullet \frac{5}{4}$
• Cancel 2 from both the numerator and denominator: $\frac{1}{3} \bullet \frac{5}{2}=\frac{5}{6}$
• Ex. After (multiply, then reduce): $\frac{2}{3}\bullet \frac{5}{4}$
• Multiply across: $\frac{2}{3}\bullet \frac{5}{4}=\frac{2\; \bullet \; 5}{3\; \bullet \; 4}=\frac{10}{12}$
• Reduce (divide by GCF): $\frac{10}{12}\div \frac{2}{2}=\frac{5}{6}$
Dividing Fractions
• Reciprocal Rule
• For any nonzero numbers a and b, the fractions $\frac{a}{b}$ and $\frac{b}{a}$ are reciprocals.
• The product of reciprocals is 1.
• Ex. $\frac{2}{3}\bullet \frac{3}{2}=\frac{1}{1}=1$
• Keep –>  Change –>  Flip
1. Keep the first fraction as is
2. Change the sign to multiplication
3. Flip (take the reciprocal of) the second fraction
• Ex. $\frac{4}{5}\div \frac{6}{7}=\frac{4}{5}\bullet \frac{7}{6}$
• Multiply, then reduce; or, you can reduce, then multiply (either way works)
• Multiply, then reduce: $\frac{4}{5}\bullet \frac{7}{6}=\frac{4\; \bullet \; 7}{5\; \bullet \; 6}=\frac{28}{30}$
• Reduce (divide by GCF): $\frac{28}{30}\div \frac{2}{2}=\frac{14}{15}$

## Notes

Multiplication of Rational Expressions
• Multiply rational expressions as if they are fractions.
• In math rules, we use lowercase variables to stand for numbers and capital letters to stand for expressions.
• Expressions can be numbers, variables, or polynomials.

• Ex: $\frac{1}{x}\bullet \frac{2}{x+1}$
• Multiply across: $\frac{1\; \bullet \; 2}{x\; \bullet \; (x+1)}=\frac{2}{x^{2}+x}$
• More Examples

Example 3

Division of Rational Expressions
• Divide rational expressions as if they are fractions.
• If If $\frac{A}{B}$ and $\frac{C}{D}$ are rational expressions, then $\frac{A}{B}\; \div \frac{C}{D}=\frac{A\; \bullet \; D}{B\; \bullet \; C}$
• Ex. $\; \frac{2}{x}\div \frac{x}{x+1}$
• Keep –> Change –> Flip: $\frac{2}{x}\bullet \frac{x+1}{x}$
• Multiply across: $\frac{2\; \bullet \; (x+1)}{x\; \bullet \; x}$
• Simplify: $\frac{2x+2}{x^{2}}$
• More Examples

• You must simplify the expressions if they can be simplified.