Integer Exponents Worksheet featuring ten math problems to evaluate expressions with integer exponents.

Integer Exponents Worksheet Solution

We will solve each expression step by step using the rules of exponents. The key rules we will use are:
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} $
5. $ a^0 = 1 $ (for $ a \neq 0 $)

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#### Problem 1: $(5^2 \cdot 5^8) \div (5^3)^{-2}$

1. Simplify the numerator:
$$
5^2 \cdot 5^8 = 5^{2+8} = 5^{10}
$$

2. Simplify the denominator:
$$
(5^3)^{-2} = 5^{3 \cdot (-2)} = 5^{-6}
$$

3. Divide the two results:
$$
\frac{5^{10}}{5^{-6}} = 5^{10 - (-6)} = 5^{10 + 6} = 5^{16}
$$

Answer:
$$
\boxed{5^{16}}
$$

---

#### Problem 2: $(-2)^3 \cdot (-2)^1 \cdot (-2)^2 \cdot (-2)^6$

1. Use the rule $ a^m \cdot a^n = a^{m+n} $ to combine the exponents:
$$
(-2)^3 \cdot (-2)^1 \cdot (-2)^2 \cdot (-2)^6 = (-2)^{3+1+2+6} = (-2)^{12}
$$

2. Since the exponent is even, $ (-2)^{12} $ is positive:
$$
(-2)^{12} = 2^{12}
$$

Answer:
$$
\boxed{2^{12}}
$$

---

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

1. Simplify the first term:
$$
(3^3 \cdot 3^1)^2 = (3^{3+1})^2 = (3^4)^2 = 3^{4 \cdot 2} = 3^8
$$

2. Simplify the second term:
$$
\frac{(-2)^5}{(-2)^2} = (-2)^{5-2} = (-2)^3
$$
Raise this result to the power of 3:
$$
\left[ (-2)^3 \right]^3 = (-2)^{3 \cdot 3} = (-2)^9
$$

3. Combine the results:
$$
3^8 + (-2)^9
$$

Since $ 3^8 $ and $ (-2)^9 $ are not like terms, the expression is already in its simplest form.

Answer:
$$
\boxed{3^8 + (-2)^9}
$$

---

#### Problem 4: $(-9)^7 \div (-9)^6$

1. Use the rule $ \frac{a^m}{a^n} = a^{m-n} $:
$$
\frac{(-9)^7}{(-9)^6} = (-9)^{7-6} = (-9)^1 = -9
$$

Answer:
$$
\boxed{-9}
$$

---

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

1. Simplify the numerator:
$$
3^2 \cdot 3^{-1} = 3^{2 + (-1)} = 3^1 = 3
$$
Raise this result to the power of 2:
$$
(3^1)^2 = 3^2
$$

2. Divide by 3:
$$
\frac{3^2}{3} = 3^{2-1} = 3^1 = 3
$$

Answer:
$$
\boxed{3}
$$

---

#### Problem 6: $\frac{(3^{-2})^2}{4^2}$

1. Simplify the numerator:
$$
(3^{-2})^2 = 3^{-2 \cdot 2} = 3^{-4}
$$

2. Simplify the denominator:
$$
4^2 = 16
$$

3. Combine the results:
$$
\frac{3^{-4}}{16} = \frac{1}{3^4} \cdot \frac{1}{16} = \frac{1}{81 \cdot 16} = \frac{1}{1296}
$$

Answer:
$$
\boxed{\frac{1}{1296}}
$$

---

#### Problem 7: $\left( \frac{3^2 \cdot 3^3}{2^3 \cdot 3^{-1}} \right)^{-1}$

1. Simplify the numerator:
$$
3^2 \cdot 3^3 = 3^{2+3} = 3^5
$$

2. Simplify the denominator:
$$
2^3 \cdot 3^{-1} = 8 \cdot 3^{-1} = \frac{8}{3}
$$

3. Combine the fraction:
$$
\frac{3^5}{\frac{8}{3}} = 3^5 \cdot \frac{3}{8} = \frac{3^6}{8}
$$

4. Take the reciprocal (since the entire expression is raised to the power of $-1$):
$$
\left( \frac{3^6}{8} \right)^{-1} = \frac{8}{3^6}
$$

Answer:
$$
\boxed{\frac{8}{3^6}}
$$

---

#### Problem 8: $\frac{(-3)^4 - 2^5}{7^2} \cdot 2^3$

1. Simplify the numerator:
$$
(-3)^4 = 81 \quad \text{and} \quad 2^5 = 32
$$
$$
(-3)^4 - 2^5 = 81 - 32 = 49
$$

2. Simplify the denominator:
$$
7^2 = 49
$$

3. Simplify the fraction:
$$
\frac{49}{49} = 1
$$

4. Multiply by $ 2^3 $:
$$
1 \cdot 2^3 = 8
$$

Answer:
$$
\boxed{8}
$$

---

#### Problem 9: $(-3)^3 \div 3 \cdot 2^4$

1. Simplify $ (-3)^3 $:
$$
(-3)^3 = -27
$$

2. Divide by 3:
$$
\frac{-27}{3} = -9
$$

3. Multiply by $ 2^4 $:
$$
-9 \cdot 2^4 = -9 \cdot 16 = -144
$$

Answer:
$$
\boxed{-144}
$$

---

#### Problem 10: $(-7)^0 \div 8 \cdot (-2)^3$

1. Simplify $ (-7)^0 $:
$$
(-7)^0 = 1
$$

2. Divide by 8:
$$
\frac{1}{8}
$$

3. Multiply by $ (-2)^3 $:
$$
(-2)^3 = -8
$$
$$
\frac{1}{8} \cdot (-8) = -1
$$

Answer:
$$
\boxed{-1}
$$

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Final Answers:

1. $\boxed{5^{16}}$
2. $\boxed{2^{12}}$
3. $\boxed{3^8 + (-2)^9}$
4. $\boxed{-9}$
5. $\boxed{3}$
6. $\boxed{\frac{1}{1296}}$
7. $\boxed{\frac{8}{3^6}}$
8. $\boxed{8}$
9. $\boxed{-144}$
10. $\boxed{-1}$