Math worksheet focusing on the quotient rule and negative exponents, designed for practice and learning.

Let’s solve each problem step by step using the Quotient Rule and rules for Negative Exponents.

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Key Rules to Remember:

- Quotient Rule for Exponents:
\(\frac{a^m}{a^n} = a^{m-n}\) (when bases are the same)

- Negative Exponent Rule:
\(a^{-n} = \frac{1}{a^n}\) or \(\frac{1}{a^{-n}} = a^n\)

- Simplify coefficients by reducing fractions.

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## Problem 1: \(\frac{2n^2}{n}\)

- Coefficient: 2 stays as is.
- Variable: \(\frac{n^2}{n} = n^{2-1} = n^1 = n\)
✔ Answer: \(2n\)

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## Problem 2: \(\frac{8x^3}{10x^5}\)

- Simplify coefficients: \(\frac{8}{10} = \frac{4}{5}\)
- Variables: \(\frac{x^3}{x^5} = x^{3-5} = x^{-2}\)
- Apply negative exponent rule: \(x^{-2} = \frac{1}{x^2}\)
✔ Answer: \(\frac{4}{5x^2}\)

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## Problem 3: \(\frac{12x^3}{9y^8}\)

- Coefficients: \(\frac{12}{9} = \frac{4}{3}\)
- Variables: \(x^3\) in numerator, \(y^8\) in denominator — different bases, so no simplification.
✔ Answer: \(\frac{4x^3}{3y^8}\)

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## Problem 4: \(\frac{14x^4 y^7}{6x^5 y^4}\)

- Coefficients: \(\frac{14}{6} = \frac{7}{3}\)
- \(x\): \(\frac{x^4}{x^5} = x^{-1} = \frac{1}{x}\)
- \(y\): \(\frac{y^7}{y^4} = y^{3}\)
- Combine: \(\frac{7}{3} \cdot \frac{y^3}{x} = \frac{7y^3}{3x}\)
✔ Answer: \(\frac{7y^3}{3x}\)

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## Problem 5: \(\frac{11u^4}{17u^7 v^9}\)

- Coefficients: \(\frac{11}{17}\) — already reduced
- \(u\): \(\frac{u^4}{u^7} = u^{-3} = \frac{1}{u^3}\)
- \(v^9\) stays in denominator
✔ Answer: \(\frac{11}{17u^3 v^9}\)

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## Problem 6: \(\frac{4y^4}{14yx^8}\)

- Coefficients: \(\frac{4}{14} = \frac{2}{7}\)
- \(y\): \(\frac{y^4}{y} = y^{3}\)
- \(x^8\) in denominator
✔ Answer: \(\frac{2y^3}{7x^8}\)

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## Problem 7: \(\frac{12yx^4}{10yx^8}\)

- Coefficients: \(\frac{12}{10} = \frac{6}{5}\)
- \(y\): cancels out (\(y/y = 1\))
- \(x\): \(\frac{x^4}{x^8} = x^{-4} = \frac{1}{x^4}\)
✔ Answer: \(\frac{6}{5x^4}\)

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## Problem 8: \(\frac{18x^8 y^8}{10x^3}\)

- Coefficients: \(\frac{18}{10} = \frac{9}{5}\)
- \(x\): \(\frac{x^8}{x^3} = x^5\)
- \(y^8\) remains in numerator
✔ Answer: \(\frac{9x^5 y^8}{5}\)

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## Problem 9: \(\frac{5n^8}{20n^8}\)

- Coefficients: \(\frac{5}{20} = \frac{1}{4}\)
- \(n^8 / n^8 = 1\)
✔ Answer: \(\frac{1}{4}\)

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## Problem 10: \(\frac{16yx^4}{9x^8 y^2}\)

- Coefficients: \(\frac{16}{9}\) — already reduced
- \(y\): \(\frac{y}{y^2} = y^{-1} = \frac{1}{y}\)
- \(x\): \(\frac{x^4}{x^8} = x^{-4} = \frac{1}{x^4}\)
- Combine: \(\frac{16}{9} \cdot \frac{1}{x^4 y} = \frac{16}{9x^4 y}\)
✔ Answer: \(\frac{16}{9x^4 y}\)

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## ✔ Final Answers:

1) \(2n\)
2) \(\frac{4}{5x^2}\)
3) \(\frac{4x^3}{3y^8}\)
4) \(\frac{7y^3}{3x}\)
5) \(\frac{11}{17u^3 v^9}\)
6) \(\frac{2y^3}{7x^8}\)
7) \(\frac{6}{5x^4}\)
8) \(\frac{9x^5 y^8}{5}\)
9) \(\frac{1}{4}\)
10) \(\frac{16}{9x^4 y}\)

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