Exponents Worksheets with Answer Key - Free Printable

Let's solve each problem step by step using the Laws of Exponents.

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1. Simplify using product of powers rule

> Product of Powers Rule: $ a^m \cdot a^n = a^{m+n} $

#### ① $ y^{40} y^{20} $
- Same base: $ y $
- Add exponents: $ 40 + 20 = 60 $
- Answer: $ y^{60} $

#### ② $ 3x^2 \cdot 2y^4 $
- Multiply coefficients: $ 3 \cdot 2 = 6 $
- Variables are different: $ x^2 $ and $ y^4 $
- No like bases to combine
- Answer: $ 6x^2 y^4 $

#### ③ $ (3abc)(2a^2b) $
- Multiply coefficients: $ 3 \cdot 2 = 6 $
- Combine like variables:
- $ a \cdot a^2 = a^{1+2} = a^3 $
- $ b \cdot b = b^{1+1} = b^2 $
- $ c $ remains as $ c $
- Answer: $ 6a^3b^2c $

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2. Simplify using quotient of powers rule

> Quotient of Powers Rule: $ \frac{a^m}{a^n} = a^{m-n} $

#### ④ $ \frac{x^4 y^7}{x^3 y^3} $
- $ x^4 / x^3 = x^{4-3} = x^1 = x $
- $ y^7 / y^3 = y^{7-3} = y^4 $
- Answer: $ x y^4 $

#### ⑤ $ \frac{a^{10} b^{20}}{a^2 b^{10}} $
- $ a^{10}/a^2 = a^{10-2} = a^8 $
- $ b^{20}/b^{10} = b^{20-10} = b^{10} $
- Answer: $ a^8 b^{10} $

#### ⑥ $ \frac{52a^{10} b^2}{5ab^2} $
- Coefficients: $ 52/5 = \frac{52}{5} $
- $ a^{10}/a = a^{10-1} = a^9 $
- $ b^2 / b^2 = b^{2-2} = b^0 = 1 $
- So, only $ a^9 $ remains
- Answer: $ \frac{52}{5} a^9 $

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3. Simplify using negative and zero exponent rules

> Zero Exponent Rule: $ a^0 = 1 $ (for $ a \ne 0 $)
> Negative Exponent Rule: $ a^{-n} = \frac{1}{a^n} $

#### ⑦ $ (-2)^0 $
- Any nonzero number to the power 0 is 1
- Answer: $ 1 $

#### ⑧ $ (-3)^{-4} $
- Negative exponent: $ = \frac{1}{(-3)^4} $
- $ (-3)^4 = 81 $
- Answer: $ \frac{1}{81} $

#### ⑨ $ \frac{1}{4^{-2}} $
- Recall: $ \frac{1}{a^{-n}} = a^n $
- So $ \frac{1}{4^{-2}} = 4^2 = 16 $
- Answer: $ 16 $

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4. Simplify using power rule

> Power Rule: $ (a^m)^n = a^{m \cdot n} $
> Also: $ (ab)^n = a^n b^n $

#### ⑩ $ (4x^2)^6 $
- Apply power to both coefficient and variable:
- $ 4^6 = 4096 $
- $ (x^2)^6 = x^{2 \cdot 6} = x^{12} $
- Answer: $ 4096x^{12} $

#### ⑪ $ (6x^2 y^4)^2 $
- Apply square to each part:
- $ 6^2 = 36 $
- $ (x^2)^2 = x^{4} $
- $ (y^4)^2 = y^{8} $
- Answer: $ 36x^4 y^8 $

#### ⑫ $ (x^{-4})^{-4} $
- Use power rule: $ x^{-4 \cdot (-4)} = x^{16} $
- Answer: $ x^{16} $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| ① | $ y^{60} $ |
| ② | $ 6x^2 y^4 $ |
| ③ | $ 6a^3 b^2 c $ |
| ④ | $ x y^4 $ |
| ⑤ | $ a^8 b^{10} $ |
| ⑥ | $ \frac{52}{5} a^9 $ |
| ⑦ | $ 1 $ |
| ⑧ | $ \frac{1}{81} $ |
| ⑨ | $ 16 $ |
| ⑩ | $ 4096x^{12} $ |
| ⑪ | $ 36x^4 y^8 $ |
| ⑫ | $ x^{16} $ |

Let me know if you'd like this in a printable format or need explanations for any specific rule!