Exponents and Division worksheet with 12 problems to simplify expressions using positive exponents.

Let's solve each of these exponent and division problems step by step. The goal is to simplify each expression so that the answer contains only positive exponents.

We'll use the following laws of exponents:

1. $\frac{a^m}{a^n} = a^{m-n}$
2. $a^{-n} = \frac{1}{a^n}$
3. When dividing variables, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
4. For coefficients (numbers), divide normally.
5. Keep only positive exponents in the final answer.

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1) $\frac{8^a}{8}$

- $8^a$ divided by $8 = 8^1$
- So: $\frac{8^a}{8^1} = 8^{a-1}$

✔ Answer: $8^{a-1}$

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2) $\frac{8c^8}{2c^4}$

- Divide coefficients: $\frac{8}{2} = 4$
- Subtract exponents for $c$: $c^{8-4} = c^4$

✔ Answer: $4c^4$

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3) $\frac{n^7}{n}$

- $n = n^1$, so $\frac{n^7}{n^1} = n^{7-1} = n^6$

✔ Answer: $n^6$

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4) $\frac{7r^4k^3}{3r^3k^2}$

- Coefficients: $\frac{7}{3}$
- $r^{4-3} = r^1 = r$
- $k^{3-2} = k^1 = k$

✔ Answer: $\frac{7}{3}rk$

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5) $\frac{3w^7c^6}{8wc^4}$

- Coefficients: $\frac{3}{8}$
- $w^{7-1} = w^6$
- $c^{6-4} = c^2$

✔ Answer: $\frac{3}{8}w^6c^2$

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6) $\frac{8k^8}{5k}$

- Coefficients: $\frac{8}{5}$
- $k^{8-1} = k^7$

✔ Answer: $\frac{8}{5}k^7$

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7) $\frac{9g^4s^3}{3gs^2}$

- Coefficients: $\frac{9}{3} = 3$
- $g^{4-1} = g^3$
- $s^{3-2} = s^1 = s$

✔ Answer: $3g^3s$

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8) $\frac{zg}{5z^4g^8}$

- Coefficients: $\frac{1}{5}$
- $z^{1-4} = z^{-3}$
- $g^{1-8} = g^{-7}$

Now we have negative exponents, so move them to the denominator:

$$
\frac{1}{5z^3g^7}
$$

✔ Answer: $\frac{1}{5z^3g^7}$

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9) $\frac{8y^4g^3}{4y^2g^8}$

- Coefficients: $\frac{8}{4} = 2$
- $y^{4-2} = y^2$
- $g^{3-8} = g^{-5}$ → move to denominator

So: $2y^2 \cdot \frac{1}{g^5} = \frac{2y^2}{g^5}$

✔ Answer: $\frac{2y^2}{g^5}$

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10) $\frac{7y^3}{9y^4}$

- Coefficients: $\frac{7}{9}$
- $y^{3-4} = y^{-1} = \frac{1}{y}$

So: $\frac{7}{9y}$

✔ Answer: $\frac{7}{9y}$

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11) $\frac{7^7}{7^8}$

- $7^{7-8} = 7^{-1} = \frac{1}{7}$

✔ Answer: $\frac{1}{7}$

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12) $\frac{5g^8}{2g^3}$

- Coefficients: $\frac{5}{2}$
- $g^{8-3} = g^5$

✔ Answer: $\frac{5}{2}g^5$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $8^{a-1}$ |
| 2) | $4c^4$ |
| 3) | $n^6$ |
| 4) | $\frac{7}{3}rk$ |
| 5) | $\frac{3}{8}w^6c^2$ |
| 6) | $\frac{8}{5}k^7$ |
| 7) | $3g^3s$ |
| 8) | $\frac{1}{5z^3g^7}$ |
| 9) | $\frac{2y^2}{g^5}$ |
| 10) | $\frac{7}{9y}$ |
| 11) | $\frac{1}{7}$ |
| 12) | $\frac{5}{2}g^5$ |

All answers have only positive exponents, as required.

Let me know if you'd like this formatted as a printable worksheet or need explanations for any specific problem!