8th grade math (H) - Exponent Practice Sept 21 - Free Printable

Problem: Simplify the given expressions so that your answer contains only positive exponents.

We will solve each problem step by step, using the rules of exponents:

#### Rules of Exponents:
1. $ a^m \cdot a^n = a^{m+n} $
2. $ \frac{a^m}{a^n} = a^{m-n} $
3. $ (a^m)^n = a^{m \cdot n} $
4. $ a^{-n} = \frac{1}{a^n} $

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Problem 1: $\frac{2^3}{2^2}$

$$
\frac{2^3}{2^2} = 2^{3-2} = 2^1 = 2
$$

Answer: $\boxed{2}$

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Problem 2: $\frac{3}{4^4}$

This expression is already in its simplest form with positive exponents.

Answer: $\boxed{\frac{3}{4^4}}$

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Problem 3: $\frac{4^0}{4^4}$

Using the rule $ a^0 = 1 $:
$$
\frac{4^0}{4^4} = \frac{1}{4^4}
$$

Answer: $\boxed{\frac{1}{4^4}}$

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Problem 4: $\frac{3^0}{3^4}$

Using the rule $ a^0 = 1 $:
$$
\frac{3^0}{3^4} = \frac{1}{3^4}
$$

Answer: $\boxed{\frac{1}{3^4}}$

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Problem 5: $\frac{2^4}{2^0}$

Using the rule $ a^0 = 1 $:
$$
\frac{2^4}{2^0} = \frac{2^4}{1} = 2^4
$$

Answer: $\boxed{2^4}$

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Problem 6: $\frac{4a}{ab^3}$

Simplify by canceling common factors:
$$
\frac{4a}{ab^3} = \frac{4}{b^3}
$$

Answer: $\boxed{\frac{4}{b^3}}$

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Problem 7: $\frac{4x^{-1}}{x^3y^0}$

Using $ y^0 = 1 $ and $ x^{-1} = \frac{1}{x} $:
$$
\frac{4x^{-1}}{x^3y^0} = \frac{4 \cdot \frac{1}{x}}{x^3 \cdot 1} = \frac{4}{x \cdot x^3} = \frac{4}{x^{1+3}} = \frac{4}{x^4}
$$

Answer: $\boxed{\frac{4}{x^4}}$

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Problem 8: $\frac{4x^2}{2x^4}$

Simplify by dividing coefficients and subtracting exponents:
$$
\frac{4x^2}{2x^4} = \frac{4}{2} \cdot \frac{x^2}{x^4} = 2 \cdot x^{2-4} = 2 \cdot x^{-2} = \frac{2}{x^2}
$$

Answer: $\boxed{\frac{2}{x^2}}$

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Problem 9: $\frac{4m^4}{m^{-3}n^2}$

Using $ m^{-3} = \frac{1}{m^3} $:
$$
\frac{4m^4}{m^{-3}n^2} = \frac{4m^4}{\frac{1}{m^3} \cdot n^2} = 4m^4 \cdot m^3 \cdot \frac{1}{n^2} = 4m^{4+3} \cdot \frac{1}{n^2} = \frac{4m^7}{n^2}
$$

Answer: $\boxed{\frac{4m^7}{n^2}}$

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Problem 10: $\frac{m^{-2}n^{-3}}{2m^2}$

Using $ m^{-2} = \frac{1}{m^2} $ and $ n^{-3} = \frac{1}{n^3} $:
$$
\frac{m^{-2}n^{-3}}{2m^2} = \frac{\frac{1}{m^2} \cdot \frac{1}{n^3}}{2m^2} = \frac{1}{m^2 \cdot n^3 \cdot 2m^2} = \frac{1}{2m^{2+2}n^3} = \frac{1}{2m^4n^3}
$$

Answer: $\boxed{\frac{1}{2m^4n^3}}$

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Problem 11: $\frac{2x^6}{3x^4y^2}$

Simplify by subtracting exponents for like bases:
$$
\frac{2x^6}{3x^4y^2} = \frac{2}{3} \cdot \frac{x^6}{x^4} \cdot \frac{1}{y^2} = \frac{2}{3} \cdot x^{6-4} \cdot \frac{1}{y^2} = \frac{2x^2}{3y^2}
$$

Answer: $\boxed{\frac{2x^2}{3y^2}}$

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Problem 12: $\frac{3x^2y^3}{3x^2y^2}$

Simplify by canceling common factors:
$$
\frac{3x^2y^3}{3x^2y^2} = \frac{3}{3} \cdot \frac{x^2}{x^2} \cdot \frac{y^3}{y^2} = 1 \cdot 1 \cdot y^{3-2} = y
$$

Answer: $\boxed{y}$

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Problem 13: $\frac{3x^4y^4}{4x^4y^4}$

Simplify by canceling common factors:
$$
\frac{3x^4y^4}{4x^4y^4} = \frac{3}{4}
$$

Answer: $\boxed{\frac{3}{4}}$

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**Problem 14: $\frac{4x^{-4}}{4y^{-1}}$

Using $ x^{-4} = \frac{1}{x^4} $ and $ y^{-1} = \frac{1}{y} $:
$$
\frac{4x^{-4}}{4y^{-1}} = \frac{4 \cdot \frac{1}{x^4}}{4 \cdot \frac{1}{y}} = \frac{4}{4} \cdot \frac{y}{x^4} = \frac{y}{x^4}
$$

Answer: $\boxed{\frac{y}{x^4}}$

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Problem 15: $\frac{4u^4v^{-1}}{4v}$

Using $ v^{-1} = \frac{1}{v} $:
$$
\frac{4u^4v^{-1}}{4v} = \frac{4u^4 \cdot \frac{1}{v}}{4v} = \frac{4u^4}{4v \cdot v} = \frac{u^4}{v^2}
$$

Answer: $\boxed{\frac{u^4}{v^2}}$

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**Problem 16: $\frac{2x^4}{2x^4y^{-4}}$

Using $ y^{-4} = \frac{1}{y^4} $:
$$
\frac{2x^4}{2x^4y^{-4}} = \frac{2x^4}{2x^4 \cdot \frac{1}{y^4}} = \frac{2x^4 \cdot y^4}{2x^4} = y^4
$$

Answer: $\boxed{y^4}$

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**Problem 17: $\frac{4a^{-1}v^2}{4a^{-1}v^{-1} \cdot u^{-4}}$

Using $ a^{-1} = \frac{1}{a} $, $ v^{-1} = \frac{1}{v} $, and $ u^{-4} = \frac{1}{u^4} $:
$$
\frac{4a^{-1}v^2}{4a^{-1}v^{-1} \cdot u^{-4}} = \frac{4 \cdot \frac{1}{a} \cdot v^2}{4 \cdot \frac{1}{a} \cdot \frac{1}{v} \cdot \frac{1}{u^4}}
$$
Cancel common factors:
$$
= \frac{v^2}{\frac{1}{v} \cdot \frac{1}{u^4}} = v^2 \cdot v \cdot u^4 = v^{2+1} \cdot u^4 = v^3u^4
$$

Answer: $\boxed{v^3u^4}$

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**Problem 18: $\frac{x^{-2}y^2 - 3x^2y^2}{4yx^2}$

Split the fraction:
$$
\frac{x^{-2}y^2 - 3x^2y^2}{4yx^2} = \frac{x^{-2}y^2}{4yx^2} - \frac{3x^2y^2}{4yx^2}
$$
Simplify each term:
1. For $\frac{x^{-2}y^2}{4yx^2}$:
$$
\frac{x^{-2}y^2}{4yx^2} = \frac{y^2}{4yx^2 \cdot x^2} = \frac{y^2}{4yx^4} = \frac{y}{4x^4}
$$
2. For $\frac{3x^2y^2}{4yx^2}$:
$$
\frac{3x^2y^2}{4yx^2} = \frac{3y^2}{4y} = \frac{3y}{4}
$$
Combine results:
$$
\frac{y}{4x^4} - \frac{3y}{4} = \frac{y}{4x^4} - \frac{3yx^4}{4x^4} = \frac{y - 3yx^4}{4x^4}
$$

Answer: $\boxed{\frac{y - 3yx^4}{4x^4}}$

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**Problem 19: $\frac{4a^2b^2}{2a^2b^2 \cdot 2b^4}$

Simplify by canceling common factors:
$$
\frac{4a^2b^2}{2a^2b^2 \cdot 2b^4} = \frac{4}{2 \cdot 2} \cdot \frac{a^2}{a^2} \cdot \frac{b^2}{b^2 \cdot b^4} = \frac{4}{4} \cdot 1 \cdot \frac{1}{b^{2+4}} = \frac{1}{b^6}
$$

Answer: $\boxed{\frac{1}{b^6}}$

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**Problem 20: $\frac{4x^4y^6}{2x^4y^2 \cdot x^2y^2}$

Simplify by combining exponents:
$$
\frac{4x^4y^6}{2x^4y^2 \cdot x^2y^2} = \frac{4x^4y^6}{2x^{4+2}y^{2+2}} = \frac{4x^4y^6}{2x^6y^4}
$$
Simplify coefficients and subtract exponents:
$$
= \frac{4}{2} \cdot \frac{x^4}{x^6} \cdot \frac{y^6}{y^4} = 2 \cdot x^{4-6} \cdot y^{6-4} = 2 \cdot x^{-2} \cdot y^2 = \frac{2y^2}{x^2}
$$

Answer: $\boxed{\frac{2y^2}{x^2}}$

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Final Answers:

1. $\boxed{2}$
2. $\boxed{\frac{3}{4^4}}$
3. $\boxed{\frac{1}{4^4}}$
4. $\boxed{\frac{1}{3^4}}$
5. $\boxed{2^4}$
6. $\boxed{\frac{4}{b^3}}$
7. $\boxed{\frac{4}{x^4}}$
8. $\boxed{\frac{2}{x^2}}$
9. $\boxed{\frac{4m^7}{n^2}}$
10. $\boxed{\frac{1}{2m^4n^3}}$
11. $\boxed{\frac{2x^2}{3y^2}}$
12. $\boxed{y}$
13. $\boxed{\frac{3}{4}}$
14. $\boxed{\frac{y}{x^4}}$
15. $\boxed{\frac{u^4}{v^2}}$
16. $\boxed{y^4}$
17. $\boxed{v^3u^4}$
18. $\boxed{\frac{y - 3yx^4}{4x^4}}$
19. $\boxed{\frac{1}{b^6}}$
20. $\boxed{\frac{2y^2}{x^2}}$