Dividing Rational Expressions - Kuta Software - Free Printable

You've uploaded a worksheet titled “Dividing Rational Expressions” from Kuta Software — Infinite Algebra 1. The task is to simplify each expression by performing division of rational expressions.

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🔁 General Rule for Dividing Rational Expressions:

To divide two rational expressions, multiply the first expression by the reciprocal of the second:

> \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}\)

Then simplify by factoring and canceling common factors.

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Let’s solve each problem step-by-step.

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## ✔ Problem 1:
\(\frac{10n}{9} \div \frac{13n^2}{16}\)

→ Multiply by reciprocal:
\(\frac{10n}{9} \cdot \frac{16}{13n^2}\)

→ Multiply numerators and denominators:
\(\frac{10n \cdot 16}{9 \cdot 13n^2} = \frac{160n}{117n^2}\)

→ Cancel one \(n\):
\(\frac{160}{117n}\)

✔ Answer: \(\boxed{\frac{160}{117n}}\)

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## ✔ Problem 2:
\(\frac{16r}{17} \div \frac{8r}{6}\)

→ Multiply by reciprocal:
\(\frac{16r}{17} \cdot \frac{6}{8r}\)

→ Simplify before multiplying:
\(\frac{16r \cdot 6}{17 \cdot 8r} = \frac{96r}{136r}\)

→ Cancel \(r\) and reduce fraction:
\(\frac{96}{136} = \frac{12}{17}\) (divide numerator and denominator by 8)

✔ Answer: \(\boxed{\frac{12}{17}}\)

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## ✔ Problem 3:
\(\frac{2}{7} \div \frac{18}{8x^3}\)

→ Multiply by reciprocal:
\(\frac{2}{7} \cdot \frac{8x^3}{18}\)

→ Multiply:
\(\frac{16x^3}{126}\)

→ Simplify: divide numerator and denominator by 2 → \(\frac{8x^3}{63}\)

✔ Answer: \(\boxed{\frac{8x^3}{63}}\)

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## ✔ Problem 4:
\(\frac{12}{7} \div \frac{4}{11r}\)

→ Multiply by reciprocal:
\(\frac{12}{7} \cdot \frac{11r}{4}\)

→ Simplify: 12 and 4 → 3 and 1
\(\frac{3 \cdot 11r}{7} = \frac{33r}{7}\)

✔ Answer: \(\boxed{\frac{33r}{7}}\)

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## ✔ Problem 5:
\(\frac{7}{18} \div \frac{6}{9a}\)

→ Multiply by reciprocal:
\(\frac{7}{18} \cdot \frac{9a}{6}\)

→ Simplify: 9 and 18 → 1 and 2; 6 and 7 no common factors
\(\frac{7 \cdot 9a}{18 \cdot 6} = \frac{63a}{108}\)

→ Reduce: divide numerator and denominator by 9 → \(\frac{7a}{12}\)

✔ Answer: \(\boxed{\frac{7a}{12}}\)

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## ✔ Problem 6:
\(\frac{5}{20} \div \frac{5x}{3}\)

→ Multiply by reciprocal:
\(\frac{5}{20} \cdot \frac{3}{5x}\)

→ Simplify: 5 cancels top and bottom
\(\frac{1}{20} \cdot \frac{3}{x} = \frac{3}{20x}\)

✔ Answer: \(\boxed{\frac{3}{20x}}\)

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## ✔ Problem 7:
\(\frac{4n}{n - 6} \div \frac{4n}{8n - 48}\)

→ First, factor denominator in second expression:
\(8n - 48 = 8(n - 6)\)

So:
\(\frac{4n}{n - 6} \div \frac{4n}{8(n - 6)}\)

→ Multiply by reciprocal:
\(\frac{4n}{n - 6} \cdot \frac{8(n - 6)}{4n}\)

→ Cancel \(4n\) and \((n - 6)\):
Left with \(8\)

✔ Answer: \(\boxed{8}\)

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## ✔ Problem 8:
\(\frac{3}{28b} \div \frac{3}{b + 1}\)

→ Multiply by reciprocal:
\(\frac{3}{28b} \cdot \frac{b + 1}{3}\)

→ Cancel 3:
\(\frac{b + 1}{28b}\)

✔ Answer: \(\boxed{\frac{b + 1}{28b}}\)

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## ✔ Problem 9:
\(\frac{7a^2}{7a^2 + 56a} \div \frac{2}{a^2 + 7a - 8}\)

→ Factor both denominators and numerators:

First expression:
Numerator: \(7a^2\)
Denominator: \(7a(a + 8)\)

Second expression (divisor):
Numerator: 2
Denominator: \(a^2 + 7a - 8 = (a + 8)(a - 1)\)

So original expression becomes:
\(\frac{7a^2}{7a(a + 8)} \div \frac{2}{(a + 8)(a - 1)}\)

→ Multiply by reciprocal:
\(\frac{7a^2}{7a(a + 8)} \cdot \frac{(a + 8)(a - 1)}{2}\)

→ Cancel 7, one \(a\), and \((a + 8)\):
Left with: \(\frac{a(a - 1)}{2}\)

✔ Answer: \(\boxed{\frac{a(a - 1)}{2}}\)

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## ✔ Problem 10:
\(\frac{6}{28x + 4} \div \frac{6}{35x + 5}\)

→ Factor denominators:

First: \(28x + 4 = 4(7x + 1)\)
Second: \(35x + 5 = 5(7x + 1)\)

Expression becomes:
\(\frac{6}{4(7x + 1)} \div \frac{6}{5(7x + 1)}\)

→ Multiply by reciprocal:
\(\frac{6}{4(7x + 1)} \cdot \frac{5(7x + 1)}{6}\)

→ Cancel 6 and \((7x + 1)\):
Left with: \(\frac{5}{4}\)

✔ Answer: \(\boxed{\frac{5}{4}}\)

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## 📝 Final Answers Summary:

1) \(\boxed{\frac{160}{117n}}\)
2) \(\boxed{\frac{12}{17}}\)
3) \(\boxed{\frac{8x^3}{63}}\)
4) \(\boxed{\frac{33r}{7}}\)
5) \(\boxed{\frac{7a}{12}}\)
6) \(\boxed{\frac{3}{20x}}\)
7) \(\boxed{8}\)
8) \(\boxed{\frac{b + 1}{28b}}\)
9) \(\boxed{\frac{a(a - 1)}{2}}\)
10) \(\boxed{\frac{5}{4}}\)

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