Properties of Exponents - Kuta Software - Free Printable

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

We will solve each problem step by step, applying the relevant properties of exponents:

#### Properties 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} $
5. Zero Exponent Rule: $ a^0 = 1 $ (for $ a \neq 0 $)
6. Distributive Property for Powers: $ (ab)^n = a^n b^n $

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Solutions:

#### 1) $ 2m^7 \cdot 2m^4 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $.
- Combine the coefficients and the exponents:
$$
2m^7 \cdot 2m^4 = (2 \cdot 2) \cdot m^{7+4} = 4m^{11}
$$
- Answer: $ \boxed{4m^{11}} $

#### 2) $ m^8 \cdot 2m^{-3} $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $.
- Combine the coefficients and the exponents:
$$
m^8 \cdot 2m^{-3} = 2 \cdot m^{8 + (-3)} = 2m^5
$$
- Answer: $ \boxed{2m^5} $

#### 3) $ 4r^{-2} \cdot 2r^2 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $.
- Combine the coefficients and the exponents:
$$
4r^{-2} \cdot 2r^2 = (4 \cdot 2) \cdot r^{-2+2} = 8r^0
$$
- Apply the Zero Exponent Rule: $ r^0 = 1 $.
$$
8r^0 = 8 \cdot 1 = 8
$$
- Answer: $ \boxed{8} $

#### 4) $ 4n^4 \cdot 2n^{-2} $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $.
- Combine the coefficients and the exponents:
$$
4n^4 \cdot 2n^{-2} = (4 \cdot 2) \cdot n^{4 + (-2)} = 8n^2
$$
- Answer: $ \boxed{8n^2} $

#### 5) $ 2k^4 \cdot 4k $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $.
- Note that $ k = k^1 $.
- Combine the coefficients and the exponents:
$$
2k^4 \cdot 4k = (2 \cdot 4) \cdot k^{4+1} = 8k^5
$$
- Answer: $ \boxed{8k^5} $

#### 6) $ 2x^3y^{-3} \cdot 2x^{-1}y^3 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $.
- Combine the coefficients and the exponents for $ x $ and $ y $ separately:
$$
2x^3y^{-3} \cdot 2x^{-1}y^3 = (2 \cdot 2) \cdot x^{3 + (-1)} \cdot y^{-3 + 3}
$$
$$
= 4x^2 \cdot y^0
$$
- Apply the Zero Exponent Rule: $ y^0 = 1 $.
$$
4x^2 \cdot y^0 = 4x^2 \cdot 1 = 4x^2
$$
- Answer: $ \boxed{4x^2} $

#### 7) $ 2y^2 \cdot 3x $
- Combine the coefficients and write the variables in alphabetical order:
$$
2y^2 \cdot 3x = (2 \cdot 3) \cdot x \cdot y^2 = 6xy^2
$$
- Answer: $ \boxed{6xy^2} $

#### 8) $ 4v^3 \cdot v u^2 $
- Combine the coefficients and use the Product Rule for $ v $.
- Note that $ v = v^1 $.
$$
4v^3 \cdot v u^2 = 4 \cdot v^{3+1} \cdot u^2 = 4v^4u^2
$$
- Answer: $ \boxed{4v^4u^2} $

#### 9) $ 4a^3b^2 \cdot 3a^{-2}b^{-3} $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $.
- Combine the coefficients and the exponents for $ a $ and $ b $ separately:
$$
4a^3b^2 \cdot 3a^{-2}b^{-3} = (4 \cdot 3) \cdot a^{3 + (-2)} \cdot b^{2 + (-3)}
$$
$$
= 12a^1 \cdot b^{-1}
$$
- Rewrite $ b^{-1} $ using the Negative Exponent Rule: $ b^{-1} = \frac{1}{b} $.
$$
12a^1 \cdot b^{-1} = 12a \cdot \frac{1}{b} = \frac{12a}{b}
$$
- Answer: $ \boxed{\frac{12a}{b}} $

#### 10) $ x^2y^{-4} \cdot x^3y^2 $
- Use the Product Rule: $ a^m \cdot a^n = a^{m+n} $.
- Combine the coefficients and the exponents for $ x $ and $ y $ separately:
$$
x^2y^{-4} \cdot x^3y^2 = x^{2+3} \cdot y^{-4+2}
$$
$$
= x^5 \cdot y^{-2}
$$
- Rewrite $ y^{-2} $ using the Negative Exponent Rule: $ y^{-2} = \frac{1}{y^2} $.
$$
x^5 \cdot y^{-2} = x^5 \cdot \frac{1}{y^2} = \frac{x^5}{y^2}
$$
- Answer: $ \boxed{\frac{x^5}{y^2}} $

#### 11) $ (x^3)^0 $
- Apply the Zero Exponent Rule: $ a^0 = 1 $ (for $ a \neq 0 $).
$$
(x^3)^0 = 1
$$
- Answer: $ \boxed{1} $

#### 12) $ (2x^2)^{-4} $
- Use the Power Rule: $ (a^m)^n = a^{m \cdot n} $.
- Apply the Negative Exponent Rule: $ a^{-n} = \frac{1}{a^n} $.
$$
(2x^2)^{-4} = \frac{1}{(2x^2)^4}
$$
- Distribute the exponent 4 using the Distributive Property for Powers: $ (ab)^n = a^n b^n $.
$$
(2x^2)^4 = 2^4 \cdot (x^2)^4
$$
- Simplify $ 2^4 $ and apply the Power Rule to $ (x^2)^4 $:
$$
2^4 = 16, \quad (x^2)^4 = x^{2 \cdot 4} = x^8
$$
$$
(2x^2)^4 = 16x^8
$$
- Therefore:
$$
(2x^2)^{-4} = \frac{1}{16x^8}
$$
- Answer: $ \boxed{\frac{1}{16x^8}} $

#### 13) $ (4r^0)^4 $
- Apply the Zero Exponent Rule: $ r^0 = 1 $.
$$
(4r^0)^4 = (4 \cdot 1)^4 = 4^4
$$
- Simplify $ 4^4 $:
$$
4^4 = 256
$$
- Answer: $ \boxed{256} $

#### 14) $ (4u^3)^2 $
- Use the Distributive Property for Powers: $ (ab)^n = a^n b^n $.
$$
(4u^3)^2 = 4^2 \cdot (u^3)^2
$$
- Simplify $ 4^2 $ and apply the Power Rule to $ (u^3)^2 $:
$$
4^2 = 16, \quad (u^3)^2 = u^{3 \cdot 2} = u^6
$$
$$
(4u^3)^2 = 16u^6
$$
- Answer: $ \boxed{16u^6} $

#### 15) $ (3k^4)^6 $
- Use the Distributive Property for Powers: $ (ab)^n = a^n b^n $.
$$
(3k^4)^6 = 3^6 \cdot (k^4)^6
$$
- Simplify $ 3^6 $ and apply the Power Rule to $ (k^4)^6 $:
$$
3^6 = 729, \quad (k^4)^6 = k^{4 \cdot 6} = k^{24}
$$
$$
(3k^4)^6 = 729k^{24}
$$
- Answer: $ \boxed{729k^{24}} $

#### 16) $ (4xy)^{-1} $
- Use the Negative Exponent Rule: $ a^{-n} = \frac{1}{a^n} $.
$$
(4xy)^{-1} = \frac{1}{(4xy)^1} = \frac{1}{4xy}
$$
- Answer: $ \boxed{\frac{1}{4xy}} $

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Final Answers:

1. $ \boxed{4m^{11}} $
2. $ \boxed{2m^5} $
3. $ \boxed{8} $
4. $ \boxed{8n^2} $
5. $ \boxed{8k^5} $
6. $ \boxed{4x^2} $
7. $ \boxed{6xy^2} $
8. $ \boxed{4v^4u^2} $
9. $ \boxed{\frac{12a}{b}} $
10. $ \boxed{\frac{x^5}{y^2}} $
11. $ \boxed{1} $
12. $ \boxed{\frac{1}{16x^8}} $
13. $ \boxed{256} $
14. $ \boxed{16u^6} $
15. $ \boxed{729k^{24}} $
16. $ \boxed{\frac{1}{4xy}} $