Exponents Worksheets with Answer Key - Free Printable

Problem: Simplify the given expressions using the laws of exponents. Write your answers in positive exponents.

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

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Part 1: Simplify the Expressions

#### Problem ①: $(a^4)^2$
- Use the Power Rule: $ (a^m)^n = a^{m \cdot n} $
- Here, $ m = 4 $ and $ n = 2 $:
$$
(a^4)^2 = a^{4 \cdot 2} = a^8
$$

Answer: $\boxed{a^8}$

#### Problem ②: $ p^6 \cdot p^{14} $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $
- Here, $ m = 6 $ and $ n = 14 $:
$$
p^6 \cdot p^{14} = p^{6+14} = p^{20}
$$

Answer: $\boxed{p^{20}}$

#### Problem ③: $ \frac{p^7}{p^5} $
- Use the Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
- Here, $ m = 7 $ and $ n = 5 $:
$$
\frac{p^7}{p^5} = p^{7-5} = p^2
$$

Answer: $\boxed{p^2}$

#### Problem ④: $ (z^2)^3 $
- Use the Power Rule: $ (a^m)^n = a^{m \cdot n} $
- Here, $ m = 2 $ and $ n = 3 $:
$$
(z^2)^3 = z^{2 \cdot 3} = z^6
$$

Answer: $\boxed{z^6}$

#### Problem ⑤: $ \frac{q^{10}}{q^6} $
- Use the Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
- Here, $ m = 10 $ and $ n = 6 $:
$$
\frac{q^{10}}{q^6} = q^{10-6} = q^4
$$

Answer: $\boxed{q^4}$

#### Problem ⑥: $ \frac{l^2}{l} $
- Use the Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
- Here, $ m = 2 $ and $ n = 1 $:
$$
\frac{l^2}{l} = l^{2-1} = l^1 = l
$$

Answer: $\boxed{l}$

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Part 2: Simplify Using the Laws of Exponents

#### Problem ⑦: $ (x^3b)^4(xb^6)^2 $
- First, apply the Power Rule to each term inside the parentheses:
$$
(x^3b)^4 = (x^3)^4 \cdot b^4 = x^{3 \cdot 4} \cdot b^4 = x^{12}b^4
$$
$$
(xb^6)^2 = (x)^2 \cdot (b^6)^2 = x^2 \cdot b^{6 \cdot 2} = x^2b^{12}
$$
- Now, multiply the results:
$$
(x^3b)^4(xb^6)^2 = (x^{12}b^4)(x^2b^{12})
$$
- Use the Product Rule for both $ x $ and $ b $:
$$
x^{12} \cdot x^2 = x^{12+2} = x^{14}
$$
$$
b^4 \cdot b^{12} = b^{4+12} = b^{16}
$$
- Combine the results:
$$
(x^3b)^4(xb^6)^2 = x^{14}b^{16}
$$

Answer: $\boxed{x^{14}b^{16}}$

#### Problem ⑧: $ \left( \frac{a^2b}{b^{-3}c^4} \right)^3 (a^{-3}b)^{-2} $
- Simplify the fraction inside the first term:
$$
\frac{a^2b}{b^{-3}c^4} = a^2b \cdot \frac{1}{b^{-3}} \cdot \frac{1}{c^4}
$$
- Use the Negative Exponent Rule: $ \frac{1}{b^{-3}} = b^3 $
$$
\frac{a^2b}{b^{-3}c^4} = a^2b \cdot b^3 \cdot \frac{1}{c^4} = a^2b^{1+3} \cdot c^{-4} = a^2b^4c^{-4}
$$
- Raise the entire expression to the power of 3:
$$
\left( a^2b^4c^{-4} \right)^3 = (a^2)^3 \cdot (b^4)^3 \cdot (c^{-4})^3
$$
- Apply the Power Rule:
$$
(a^2)^3 = a^{2 \cdot 3} = a^6, \quad (b^4)^3 = b^{4 \cdot 3} = b^{12}, \quad (c^{-4})^3 = c^{-4 \cdot 3} = c^{-12}
$$
$$
\left( a^2b^4c^{-4} \right)^3 = a^6b^{12}c^{-12}
$$
- Simplify the second term $ (a^{-3}b)^{-2} $:
$$
(a^{-3}b)^{-2} = (a^{-3})^{-2} \cdot (b)^{-2}
$$
- Apply the Power Rule:
$$
(a^{-3})^{-2} = a^{-3 \cdot -2} = a^6, \quad (b)^{-2} = b^{-2}
$$
$$
(a^{-3}b)^{-2} = a^6b^{-2}
$$
- Multiply the two results:
$$
\left( a^6b^{12}c^{-12} \right) \cdot (a^6b^{-2}) = a^6 \cdot a^6 \cdot b^{12} \cdot b^{-2} \cdot c^{-12}
$$
- Use the Product Rule:
$$
a^6 \cdot a^6 = a^{6+6} = a^{12}, \quad b^{12} \cdot b^{-2} = b^{12-2} = b^{10}, \quad c^{-12} = c^{-12}
$$
$$
\left( a^6b^{12}c^{-12} \right) \cdot (a^6b^{-2}) = a^{12}b^{10}c^{-12}
$$
- Write the final answer with positive exponents:
$$
a^{12}b^{10}c^{-12} = \frac{a^{12}b^{10}}{c^{12}}
$$

Answer: $\boxed{\frac{a^{12}b^{10}}{c^{12}}}$

#### Problem ⑨: $ \left( \frac{x^2y^7}{x^{-2}y^4} \right)^2 $
- Simplify the fraction inside the parentheses:
$$
\frac{x^2y^7}{x^{-2}y^4} = x^2 \cdot \frac{1}{x^{-2}} \cdot y^7 \cdot \frac{1}{y^4}
$$
- Use the Negative Exponent Rule: $ \frac{1}{x^{-2}} = x^2 $
$$
\frac{x^2y^7}{x^{-2}y^4} = x^2 \cdot x^2 \cdot y^{7-4} = x^{2+2} \cdot y^3 = x^4y^3
$$
- Raise the entire expression to the power of 2:
$$
\left( x^4y^3 \right)^2 = (x^4)^2 \cdot (y^3)^2
$$
- Apply the Power Rule:
$$
(x^4)^2 = x^{4 \cdot 2} = x^8, \quad (y^3)^2 = y^{3 \cdot 2} = y^6
$$
$$
\left( x^4y^3 \right)^2 = x^8y^6
$$

Answer: $\boxed{x^8y^6}$

#### Problem ⑩: $ \left( \frac{p^3q^5}{r^7} \right) \cdot \left( \frac{p^2r^0q^3}{p^4r^2} \right) $
- Simplify each fraction separately:
- First fraction: $ \frac{p^3q^5}{r^7} $
- Second fraction: $ \frac{p^2r^0q^3}{p^4r^2} $
- Note that $ r^0 = 1 $:
$$
\frac{p^2r^0q^3}{p^4r^2} = \frac{p^2 \cdot 1 \cdot q^3}{p^4 \cdot r^2} = \frac{p^2q^3}{p^4r^2}
$$
- Simplify using the Quotient Rule:
$$
\frac{p^2q^3}{p^4r^2} = p^{2-4} \cdot q^3 \cdot r^{-2} = p^{-2}q^3r^{-2}
$$
- Multiply the two fractions:
$$
\left( \frac{p^3q^5}{r^7} \right) \cdot \left( \frac{p^2r^0q^3}{p^4r^2} \right) = \left( p^3q^5r^{-7} \right) \cdot \left( p^{-2}q^3r^{-2} \right)
$$
- Use the Product Rule for $ p $, $ q $, and $ r $:
$$
p^3 \cdot p^{-2} = p^{3-2} = p^1 = p
$$
$$
q^5 \cdot q^3 = q^{5+3} = q^8
$$
$$
r^{-7} \cdot r^{-2} = r^{-7-2} = r^{-9}
$$
$$
\left( p^3q^5r^{-7} \right) \cdot \left( p^{-2}q^3r^{-2} \right) = pq^8r^{-9}
$$
- Write the final answer with positive exponents:
$$
pq^8r^{-9} = \frac{pq^8}{r^9}
$$

Answer: $\boxed{\frac{pq^8}{r^9}}$

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Final Answers:

1. $\boxed{a^8}$
2. $\boxed{p^{20}}$
3. $\boxed{p^2}$
4. $\boxed{z^6}$
5. $\boxed{q^4}$
6. $\boxed{l}$
7. $\boxed{x^{14}b^{16}}$
8. $\boxed{\frac{a^{12}b^{10}}{c^{12}}}$
9. $\boxed{x^8y^6}$
10. $\boxed{\frac{pq^8}{r^9}}$