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7.4 – Multiplying and Dividing Rational Expressions

Key Terms

  • No key terms for this section.


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}



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.

Alg2B 7.4 Fractions Expressions

  • 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

Alg2B 7.4 Rational Exp Ex

Example 3

Alg2B 7.4 Rational Exp Ex2


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

Alg2B 7.4 Rational Exp Div01

Alg2B 7.4 Rational Exp Div02

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

Alg2B 7.4 Rational Exp Simplified

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