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

Let's solve each expression step by step using the laws of exponents. We'll apply rules like:

- $ a^m \cdot a^n = a^{m+n} $
- $ \frac{a^m}{a^n} = a^{m-n} $
- $ (a^m)^n = a^{m \cdot n} $
- $ a^{-n} = \frac{1}{a^n} $
- $ a^0 = 1 $ (for $ a \neq 0 $)

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1. $ (5^2 \cdot 5^8) \div (5^3)^2 $

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

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

Step 3: Divide:
$ \frac{5^{10}}{5^6} = 5^{10-6} = 5^4 = 625 $

✔ Answer: $ \boxed{625} $

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2. $ (-2)^3 \cdot (-2)^1 \cdot (-2)^2 \cdot (-2)^6 $

All bases are $-2$, so add exponents:

$ (-2)^{3+1+2+6} = (-2)^{12} $

Now, $ (-2)^{12} = 2^{12} $ since even power → positive

$ 2^{12} = 4096 $

✔ Answer: $ \boxed{4096} $

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3. $ (3^3 \cdot 3)^2 + [(-2)^5 \div (-2)^2]^3 $

Part A: $ (3^3 \cdot 3)^2 $

$ 3^3 \cdot 3 = 3^{3+1} = 3^4 $
Then $ (3^4)^2 = 3^{8} = 6561 $

Part B: $ [(-2)^5 \div (-2)^2]^3 $

$ (-2)^5 \div (-2)^2 = (-2)^{5-2} = (-2)^3 = -8 $

Now raise to 3rd power: $ (-8)^3 = -512 $

Add both parts:
$ 6561 + (-512) = 6561 - 512 = 6049 $

✔ Answer: $ \boxed{6049} $

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4. $ (-9)^7 \div (-9)^6 $

$ (-9)^{7-6} = (-9)^1 = -9 $

✔ Answer: $ \boxed{-9} $

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5. $ \frac{(3^2 \cdot 3^{-1})^2}{3} $

Numerator: $ (3^2 \cdot 3^{-1})^2 $

First: $ 3^2 \cdot 3^{-1} = 3^{2-1} = 3^1 = 3 $

Then square: $ 3^2 = 9 $

Now divide by 3: $ \frac{9}{3} = 3 $

✔ Answer: $ \boxed{3} $

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6. $ \frac{(3^{-2})^2}{4^2} $

Numerator: $ (3^{-2})^2 = 3^{-4} = \frac{1}{3^4} = \frac{1}{81} $

Denominator: $ 4^2 = 16 $

So: $ \frac{1/81}{16} = \frac{1}{81 \cdot 16} = \frac{1}{1296} $

✔ Answer: $ \boxed{\frac{1}{1296}} $

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

First simplify inside the parentheses.

Numerator: $ 3^2 \cdot 3^3 = 3^{5} $

Denominator: $ 2^3 \cdot 3^{-1} = 8 \cdot \frac{1}{3} = \frac{8}{3} $

So fraction becomes:
$ \frac{3^5}{8/3} = 3^5 \cdot \frac{3}{8} = \frac{243 \cdot 3}{8} = \frac{729}{8} $

Now take the reciprocal (because of exponent $-1$):

$ \left( \frac{729}{8} \right)^{-1} = \frac{8}{729} $

✔ Answer: $ \boxed{\frac{8}{729}} $

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

Step 1: Compute numerator:
- $ (-3)^4 = 81 $ (even power)
- $ 2^5 = 32 $
- So $ 81 - 32 = 49 $

Step 2: Denominator: $ 7^2 = 49 $

So: $ \frac{49}{49} = 1 $

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

✔ Answer: $ \boxed{8} $

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9. $ (-3)^3 \div 3 \cdot 2^4 $

Step 1: $ (-3)^3 = -27 $

Step 2: $ 2^4 = 16 $

Now compute:
$ -27 \div 3 \cdot 16 $

Order of operations: left to right
$ -27 \div 3 = -9 $
$ -9 \cdot 16 = -144 $

✔ Answer: $ \boxed{-144} $

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10. $ (-7)^0 \div 8 \cdot (-2)^3 $

Step 1: $ (-7)^0 = 1 $ (any nonzero number to power 0 is 1)

Step 2: $ (-2)^3 = -8 $

Now: $ 1 \div 8 \cdot (-8) $

Left to right:
$ 1 \div 8 = \frac{1}{8} $
$ \frac{1}{8} \cdot (-8) = -1 $

✔ Answer: $ \boxed{-1} $

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✔ Final Answers:

1. $ \boxed{625} $
2. $ \boxed{4096} $
3. $ \boxed{6049} $
4. $ \boxed{-9} $
5. $ \boxed{3} $
6. $ \boxed{\frac{1}{1296}} $
7. $ \boxed{\frac{8}{729}} $
8. $ \boxed{8} $
9. $ \boxed{-144} $
10. $ \boxed{-1} $

Let me know if you'd like this in a printable format or with more explanation on any step!