Laws of Exponents worksheet - Free Printable

Problem: Simplify each expression so that all exponents are positive, and provide the solutions.

We will simplify each expression step by step using the laws of exponents. The laws we will use include:
1. \( a^m \cdot a^n = a^{m+n} \)
2. \( \frac{a^m}{a^n} = a^{m-n} \)
3. \( (a^m)^n = a^{m \cdot n} \)
4. \( a^{-n} = \frac{1}{a^n} \)

---

#### Expression 1: \( 9c^5 \cdot 5c^{-5} \)

- Combine coefficients: \( 9 \cdot 5 = 45 \)
- Combine exponents of \( c \): \( c^5 \cdot c^{-5} = c^{5 + (-5)} = c^0 = 1 \)
- Simplified expression: \( 45 \cdot 1 = 45 \)

Solution: \( 45 \)

---

#### Expression 2: \( 2k^6g^2 \cdot 3kg^3 \)

- Combine coefficients: \( 2 \cdot 3 = 6 \)
- Combine exponents of \( k \): \( k^6 \cdot k^1 = k^{6+1} = k^7 \)
- Combine exponents of \( g \): \( g^2 \cdot g^3 = g^{2+3} = g^5 \)
- Simplified expression: \( 6k^7g^5 \)

Solution: \( 6k^7g^5 \)

---

#### Expression 3: \( 7 \cdot 7^{-5} \)

- Combine exponents of \( 7 \): \( 7^1 \cdot 7^{-5} = 7^{1 + (-5)} = 7^{-4} \)
- Convert negative exponent to positive: \( 7^{-4} = \frac{1}{7^4} \)

Solution: \( \frac{1}{7^4} \)

---

#### Expression 4: \( \frac{5h^2}{8h^4} \)

- Simplify the coefficient: \( \frac{5}{8} \)
- Simplify the exponents of \( h \): \( \frac{h^2}{h^4} = h^{2-4} = h^{-2} \)
- Convert negative exponent to positive: \( h^{-2} = \frac{1}{h^2} \)
- Simplified expression: \( \frac{5}{8h^2} \)

Solution: \( \frac{5}{8h^2} \)

---

#### Expression 5: \( \frac{5s^{-2}n^{-5}}{7sn^{-6}} \)

- Simplify the coefficient: \( \frac{5}{7} \)
- Simplify the exponents of \( s \): \( \frac{s^{-2}}{s^1} = s^{-2-1} = s^{-3} \)
- Simplify the exponents of \( n \): \( \frac{n^{-5}}{n^{-6}} = n^{-5 - (-6)} = n^{-5 + 6} = n^1 = n \)
- Convert negative exponent to positive: \( s^{-3} = \frac{1}{s^3} \)
- Simplified expression: \( \frac{5n}{7s^3} \)

Solution: \( \frac{5n}{7s^3} \)

---

#### Expression 6: \( \frac{5w^2}{9w^6y^4} \)

- Simplify the coefficient: \( \frac{5}{9} \)
- Simplify the exponents of \( w \): \( \frac{w^2}{w^6} = w^{2-6} = w^{-4} \)
- The exponent of \( y \) remains \( y^4 \) in the denominator.
- Convert negative exponent to positive: \( w^{-4} = \frac{1}{w^4} \)
- Simplified expression: \( \frac{5}{9w^4y^4} \)

Solution: \( \frac{5}{9w^4y^4} \)

---

#### Expression 7: \( \frac{6^{-3}}{6} \)

- Simplify the exponents of \( 6 \): \( \frac{6^{-3}}{6^1} = 6^{-3-1} = 6^{-4} \)
- Convert negative exponent to positive: \( 6^{-4} = \frac{1}{6^4} \)

Solution: \( \frac{1}{6^4} \)

---

#### Expression 8: \( \left( \frac{1}{n} \right)^6 \cdot \left( \frac{1}{n} \right)^3 \cdot \left( \frac{1}{n} \right)^5 \)

- Rewrite each term: \( \left( \frac{1}{n} \right)^6 = n^{-6} \), \( \left( \frac{1}{n} \right)^3 = n^{-3} \), \( \left( \frac{1}{n} \right)^5 = n^{-5} \)
- Combine exponents: \( n^{-6} \cdot n^{-3} \cdot n^{-5} = n^{-6 + (-3) + (-5)} = n^{-14} \)
- Convert negative exponent to positive: \( n^{-14} = \frac{1}{n^{14}} \)

Solution: \( \frac{1}{n^{14}} \)

---

#### Expression 9: \( bs \cdot 8b^3s^5 \)

- Combine coefficients: \( 1 \cdot 8 = 8 \)
- Combine exponents of \( b \): \( b^1 \cdot b^3 = b^{1+3} = b^4 \)
- Combine exponents of \( s \): \( s^1 \cdot s^5 = s^{1+5} = s^6 \)
- Simplified expression: \( 8b^4s^6 \)

Solution: \( 8b^4s^6 \)

---

#### Expression 10: \( \frac{6d^{-5}h^2}{5d^3h^{-3}} \)

- Simplify the coefficient: \( \frac{6}{5} \)
- Simplify the exponents of \( d \): \( \frac{d^{-5}}{d^3} = d^{-5-3} = d^{-8} \)
- Simplify the exponents of \( h \): \( \frac{h^2}{h^{-3}} = h^{2 - (-3)} = h^{2+3} = h^5 \)
- Convert negative exponent to positive: \( d^{-8} = \frac{1}{d^8} \)
- Simplified expression: \( \frac{6h^5}{5d^8} \)

Solution: \( \frac{6h^5}{5d^8} \)

---

#### Expression 11: \( \frac{2^4}{2^6} \)

- Simplify the exponents of \( 2 \): \( \frac{2^4}{2^6} = 2^{4-6} = 2^{-2} \)
- Convert negative exponent to positive: \( 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \)

Solution: \( \frac{1}{4} \)

---

#### Expression 12: \( s^2 \cdot s^{-5} \cdot s^{-4} \)

- Combine exponents of \( s \): \( s^2 \cdot s^{-5} \cdot s^{-4} = s^{2 + (-5) + (-4)} = s^{-7} \)
- Convert negative exponent to positive: \( s^{-7} = \frac{1}{s^7} \)

Solution: \( \frac{1}{s^7} \)

---

Final Answers:

1. \( 45 \)
2. \( 6k^7g^5 \)
3. \( \frac{1}{7^4} \)
4. \( \frac{5}{8h^2} \)
5. \( \frac{5n}{7s^3} \)
6. \( \frac{5}{9w^4y^4} \)
7. \( \frac{1}{6^4} \)
8. \( \frac{1}{n^{14}} \)
9. \( 8b^4s^6 \)
10. \( \frac{6h^5}{5d^8} \)
11. \( \frac{1}{4} \)
12. \( \frac{1}{s^7} \)

\boxed{45, 6k^7g^5, \frac{1}{7^4}, \frac{5}{8h^2}, \frac{5n}{7s^3}, \frac{5}{9w^4y^4}, \frac{1}{6^4}, \frac{1}{n^{14}}, 8b^4s^6, \frac{6h^5}{5d^8}, \frac{1}{4}, \frac{1}{s^7}} \)