Algebra 1 Worksheets | Exponents Worksheets - Free Printable

Problem: Simplify the given expressions. Your answer should contain only positive exponents.

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

1. Product Rule: $ a^m \cdot a^n = a^{m+n} $
2. Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
3. Power of a Power Rule: $ (a^m)^n = a^{m \cdot n} $
4. Negative Exponent Rule: $ a^{-n} = \frac{1}{a^n} $

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#### Problem 1: $ 9^4 \cdot 9^3 $

Using the product rule:
$$
9^4 \cdot 9^3 = 9^{4+3} = 9^7
$$

Answer: $ \boxed{9^7} $

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#### Problem 2: $ \left( \frac{1}{w} \right)^6 \cdot \left( \frac{1}{w} \right)^2 $

Using the product rule:
$$
\left( \frac{1}{w} \right)^6 \cdot \left( \frac{1}{w} \right)^2 = \left( \frac{1}{w} \right)^{6+2} = \left( \frac{1}{w} \right)^8
$$

Rewriting with a positive exponent:
$$
\left( \frac{1}{w} \right)^8 = w^{-8}
$$

Answer: $ \boxed{w^{-8}} $

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#### Problem 3: $ r^3 s^{-2} \cdot 8r^{-3}s^4 \cdot 4r^6 $

Combine like terms using the product rule:
$$
r^3 s^{-2} \cdot 8r^{-3}s^4 \cdot 4r^6 = (8 \cdot 4) \cdot (r^3 \cdot r^{-3} \cdot r^6) \cdot (s^{-2} \cdot s^4)
$$
$$
= 32 \cdot r^{3 + (-3) + 6} \cdot s^{-2 + 4}
$$
$$
= 32 \cdot r^{6} \cdot s^{2}
$$

Answer: $ \boxed{32r^6s^2} $

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#### Problem 4: $ 7y \cdot 2y^{-2} $

Combine coefficients and use the product rule for $ y $:
$$
7y \cdot 2y^{-2} = (7 \cdot 2) \cdot (y^1 \cdot y^{-2})
$$
$$
= 14 \cdot y^{1 + (-2)}
$$
$$
= 14 \cdot y^{-1}
$$

Rewriting with a positive exponent:
$$
14 \cdot y^{-1} = \frac{14}{y}
$$

Answer: $ \boxed{\frac{14}{y}} $

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#### Problem 5: $ \left( \frac{1}{5} \right)^1 \cdot \left( \frac{1}{5} \right)^3 \cdot \left( \frac{1}{5} \right)^2 $

Using the product rule:
$$
\left( \frac{1}{5} \right)^1 \cdot \left( \frac{1}{5} \right)^3 \cdot \left( \frac{1}{5} \right)^2 = \left( \frac{1}{5} \right)^{1+3+2}
$$
$$
= \left( \frac{1}{5} \right)^6
$$

Rewriting with a positive exponent:
$$
\left( \frac{1}{5} \right)^6 = 5^{-6}
$$

Answer: $ \boxed{5^{-6}} $

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#### Problem 6: $ 5d^2n^6 \cdot 4dn^4 $

Combine coefficients and use the product rule for $ d $ and $ n $:
$$
5d^2n^6 \cdot 4dn^4 = (5 \cdot 4) \cdot (d^2 \cdot d) \cdot (n^6 \cdot n^4)
$$
$$
= 20 \cdot d^{2+1} \cdot n^{6+4}
$$
$$
= 20 \cdot d^3 \cdot n^{10}
$$

Answer: $ \boxed{20d^3n^{10}} $

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#### Problem 7: $ 8n^{-4} \cdot 9n^6 $

Combine coefficients and use the product rule for $ n $:
$$
8n^{-4} \cdot 9n^6 = (8 \cdot 9) \cdot (n^{-4} \cdot n^6)
$$
$$
= 72 \cdot n^{-4+6}
$$
$$
= 72 \cdot n^2
$$

Answer: $ \boxed{72n^2} $

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#### Problem 8: $ \left( \frac{1}{k} \right)^3 \cdot \left( \frac{1}{k} \right)^4 \cdot \left( \frac{1}{k} \right)^5 $

Using the product rule:
$$
\left( \frac{1}{k} \right)^3 \cdot \left( \frac{1}{k} \right)^4 \cdot \left( \frac{1}{k} \right)^5 = \left( \frac{1}{k} \right)^{3+4+5}
$$
$$
= \left( \frac{1}{k} \right)^{12}
$$

Rewriting with a positive exponent:
$$
\left( \frac{1}{k} \right)^{12} = k^{-12}
$$

Answer: $ \boxed{k^{-12}} $

---

#### Problem 9: $ 5z^5 \cdot 6z^{-3} \cdot 3z^{-6} $

Combine coefficients and use the product rule for $ z $:
$$
5z^5 \cdot 6z^{-3} \cdot 3z^{-6} = (5 \cdot 6 \cdot 3) \cdot (z^5 \cdot z^{-3} \cdot z^{-6})
$$
$$
= 90 \cdot z^{5 + (-3) + (-6)}
$$
$$
= 90 \cdot z^{-4}
$$

Rewriting with a positive exponent:
$$
90 \cdot z^{-4} = \frac{90}{z^4}
$$

Answer: $ \boxed{\frac{90}{z^4}} $

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#### Problem 10: $ 9 \cdot 9^{-6} $

Using the product rule:
$$
9 \cdot 9^{-6} = 9^{1 + (-6)}
$$
$$
= 9^{-5}
$$

Rewriting with a positive exponent:
$$
9^{-5} = \frac{1}{9^5}
$$

Answer: $ \boxed{\frac{1}{9^5}} $

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#### Problem 11: $ 9b \cdot 6b^3 $

Combine coefficients and use the product rule for $ b $:
$$
9b \cdot 6b^3 = (9 \cdot 6) \cdot (b^1 \cdot b^3)
$$
$$
= 54 \cdot b^{1+3}
$$
$$
= 54 \cdot b^4
$$

Answer: $ \boxed{54b^4} $

---

#### Problem 12: $ 9s^4 \cdot 7s^{-2} $

Combine coefficients and use the product rule for $ s $:
$$
9s^4 \cdot 7s^{-2} = (9 \cdot 7) \cdot (s^4 \cdot s^{-2})
$$
$$
= 63 \cdot s^{4 + (-2)}
$$
$$
= 63 \cdot s^2
$$

Answer: $ \boxed{63s^2} $

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#### Problem 13: $ 2a^3c^6 \cdot 8a^6c^2 $

Combine coefficients and use the product rule for $ a $ and $ c $:
$$
2a^3c^6 \cdot 8a^6c^2 = (2 \cdot 8) \cdot (a^3 \cdot a^6) \cdot (c^6 \cdot c^2)
$$
$$
= 16 \cdot a^{3+6} \cdot c^{6+2}
$$
$$
= 16 \cdot a^9 \cdot c^8
$$

Answer: $ \boxed{16a^9c^8} $

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#### Problem 14: $ x^2 \cdot x^5 $

Using the product rule:
$$
x^2 \cdot x^5 = x^{2+5}
$$
$$
= x^7
$$

Answer: $ \boxed{x^7} $

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Final Answers:

1. $ \boxed{9^7} $
2. $ \boxed{w^{-8}} $
3. $ \boxed{32r^6s^2} $
4. $ \boxed{\frac{14}{y}} $
5. $ \boxed{5^{-6}} $
6. $ \boxed{20d^3n^{10}} $
7. $ \boxed{72n^2} $
8. $ \boxed{k^{-12}} $
9. $ \boxed{\frac{90}{z^4}} $
10. $ \boxed{\frac{1}{9^5}} $
11. $ \boxed{54b^4} $
12. $ \boxed{63s^2} $
13. $ \boxed{16a^9c^8} $
14. $ \boxed{x^7} $