This worksheet provides ten practice problems for students to master the skill of dividing monomials in algebra.

Let's solve each of the monomial division problems step by step. The goal is to divide one monomial by another, simplifying using the rules of exponents.

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Rules to Remember:

- When dividing powers with the same base:
$$
\frac{a^m}{a^n} = a^{m-n}
$$
- Coefficients (numbers) are divided normally.
- Variables are simplified using exponent rules.

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Now, let’s solve each problem:

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1) $ \frac{32x^4y^5}{-16x^2y^3} $

- Divide coefficients: $ \frac{32}{-16} = -2 $
- Divide variables:
- $ x^4 / x^2 = x^{4-2} = x^2 $
- $ y^5 / y^3 = y^{5-3} = y^2 $
- Final answer: $ -2x^2y^2 $

✔ Answer: $ -2x^2y^2 $

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2) $ \frac{-8xy^2}{(-2xy)^2} $

First, simplify the denominator:
- $ (-2xy)^2 = (-2)^2 \cdot x^2 \cdot y^2 = 4x^2y^2 $

Now divide:
- $ \frac{-8xy^2}{4x^2y^2} $
- Coefficients: $ \frac{-8}{4} = -2 $
- $ x / x^2 = x^{1-2} = x^{-1} = \frac{1}{x} $
- $ y^2 / y^2 = y^{0} = 1 $
- So result: $ -2 \cdot \frac{1}{x} = -\frac{2}{x} $

✔ Answer: $ -\frac{2}{x} $

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3) $ \frac{-32m^3n^2}{-4n^2} $

- Coefficients: $ \frac{-32}{-4} = 8 $
- $ m^3 $ stays (no m in denominator)
- $ n^2 / n^2 = n^{0} = 1 $
- So result: $ 8m^3 $

✔ Answer: $ 8m^3 $

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4) $ \frac{pq^2r^4}{-q^3r^2} $

- Coefficients: $ \frac{1}{-1} = -1 $
- $ p $: no match → stays
- $ q^2 / q^3 = q^{2-3} = q^{-1} = \frac{1}{q} $
- $ r^4 / r^2 = r^{4-2} = r^2 $
- Combine: $ -p \cdot \frac{1}{q} \cdot r^2 = -\frac{pr^2}{q} $

✔ Answer: $ -\frac{pr^2}{q} $

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5) $ \frac{60x^7y^6}{15x^2y^6} $

- Coefficients: $ \frac{60}{15} = 4 $
- $ x^7 / x^2 = x^{5} $
- $ y^6 / y^6 = y^0 = 1 $
- Result: $ 4x^5 $

✔ Answer: $ 4x^5 $

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6) $ \frac{16a^7b^5}{-24a^4b^3} $

- Coefficients: $ \frac{16}{-24} = -\frac{2}{3} $
- $ a^7 / a^4 = a^{3} $
- $ b^5 / b^3 = b^{2} $
- Result: $ -\frac{2}{3}a^3b^2 $

✔ Answer: $ -\frac{2}{3}a^3b^2 $

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7) $ \frac{54pq^2}{-9q^2} $

- Coefficients: $ \frac{54}{-9} = -6 $
- $ p $: no match → stays
- $ q^2 / q^2 = 1 $
- Result: $ -6p $

✔ Answer: $ -6p $

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8) $ \frac{x^6}{x^2 \cdot (-4)} $

Wait — this expression is written as $ \frac{x^6}{x^2 \cdot (-4)} $. Let's clarify:

This means:
$$
\frac{x^6}{-4x^2}
$$

- Coefficients: $ \frac{1}{-4} = -\frac{1}{4} $
- $ x^6 / x^2 = x^{4} $
- Result: $ -\frac{1}{4}x^4 $

✔ Answer: $ -\frac{1}{4}x^4 $

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9) $ \frac{75x^6y^4}{-3x^2} $

- Coefficients: $ \frac{75}{-3} = -25 $
- $ x^6 / x^2 = x^4 $
- $ y^4 $ remains (no y in denominator)
- Result: $ -25x^4y^4 $

✔ Answer: $ -25x^4y^4 $

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10) $ \frac{42g^2h^3}{-6gh^4} $

- Coefficients: $ \frac{42}{-6} = -7 $
- $ g^2 / g = g^{1} $
- $ h^3 / h^4 = h^{-1} = \frac{1}{h} $
- Result: $ -7g \cdot \frac{1}{h} = -\frac{7g}{h} $

✔ Answer: $ -\frac{7g}{h} $

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✔ Final Answers:

1) $ -2x^2y^2 $
2) $ -\frac{2}{x} $
3) $ 8m^3 $
4) $ -\frac{pr^2}{q} $
5) $ 4x^5 $
6) $ -\frac{2}{3}a^3b^2 $
7) $ -6p $
8) $ -\frac{1}{4}x^4 $
9) $ -25x^4y^4 $
10) $ -\frac{7g}{h} $

Let me know if you'd like these formatted neatly or explained further!