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Chapter 0: Problem 101

Explain how to divide rational expressions.

Short Answer

Expert verified
The procedure for dividing rational expressions starts by understanding the operation, converting the division to multiplication by taking the reciprocal of the divisor, simplifying the problem, multiplying out, and doing further simplification if necessary.

Step by step solution

01

Understanding the operation

A division problem between two rational expressions can be transformed into a multiplication problem. Given a division problem, \( \frac{a}{b} \div \frac{c}{d} \), it can be changed into \( \frac{a}{b} \times \frac{d}{c} \). This operation is called 'taking the reciprocal'. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
02

Convert to Multiplication

Take the reciprocal of the divisor (second fraction) and rewrite the division problem as a multiplication problem. Our example problem \( \frac{a}{b} \div \frac{c}{d} \) becomes \( \frac{a}{b} \times \frac{d}{c} \).
03

Simplify

Before proceeding with the multiplication, we want to simplify where possible. Essentially, we are looking to cancel out terms that appear in both the numerator and denominator of the entire expression. If there are common factors between the 'crossed' numerators and denominators, they can be eliminated.
04

Multiply

After simplifying, multiply the numerators together to get the new numerator and the denominators together to get the new denominator. As an example, if our simplified problem was \( \frac{a}{b} \times \frac{d}{c} \), the answer would be \( \frac{a \times d}{b \times c} \).
05

Check if further simplification is necessary

Check to see if the resultant fraction can be simplified further; if it can, do so. Otherwise, you have your final answer.

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Most popular questions from this chapter

Describe ways in which solving a linear inequality is different than solving a linear equation.

Explain how to add or subtract rational expressions with the same denominators.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When performing the division $$\frac{7 x}{x+3}+\frac{(x+3)^{2}}{x-5}$$ I began by dividing the numerator and the denominator by the common factor, \(x+3\).

Explain how to add rational expressions having no common factors in their denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.

Describe two ways to simplify \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\).
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