Dividing Exponents Worksheet | isgb.edu.ar - Free Printable

Let's go through each problem step by step and verify the simplifications, ensuring that all answers contain only positive exponents, as required.

We'll use the following exponent rules:

1. $\frac{a^m}{a^n} = a^{m-n}$
2. $a^{-n} = \frac{1}{a^n}$
3. $\frac{1}{a^{-n}} = a^n$
4. When dividing variables with exponents, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$

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1) $\frac{9k^{-5}}{3k^5 w^{-4}}$

- Simplify coefficients: $\frac{9}{3} = 3$
- For $k$: $k^{-5} / k^5 = k^{-5 - 5} = k^{-10}$
- For $w$: $1 / w^{-4} = w^4$ (since $w^{-4}$ in denominator becomes $w^4$ in numerator)

So:
$$
\frac{9k^{-5}}{3k^5 w^{-4}} = 3 \cdot k^{-10} \cdot w^4 = \frac{3w^4}{k^{10}}
$$

✔ Correct

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2) $\frac{hd}{3h^3 d^5}$

- Coefficients: $1/3$
- $h$: $h^1 / h^3 = h^{1-3} = h^{-2}$
- $d$: $d^1 / d^5 = d^{-4}$

So:
$$
\frac{hd}{3h^3 d^5} = \frac{1}{3} \cdot h^{-2} \cdot d^{-4} = \frac{1}{3h^2 d^4}
$$

✔ Correct

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3) $\frac{7r}{9r^6}$

- Coefficients: $7/9$
- $r$: $r^1 / r^6 = r^{-5}$

So:
$$
\frac{7r}{9r^6} = \frac{7}{9} \cdot r^{-5} = \frac{7}{9r^5}
$$

✔ Correct

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4) $\frac{7^5}{7^{-5}}$

Use rule: $\frac{a^m}{a^n} = a^{m-n}$

$$
7^{5 - (-5)} = 7^{10}
$$

✔ Correct

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5) $\frac{7^{-5}}{7}$

Note: $7 = 7^1$, so:

$$
\frac{7^{-5}}{7^1} = 7^{-5 - 1} = 7^{-6} = \frac{1}{7^6}
$$

✔ Correct

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6) $\frac{3kr^3}{5k^5 r^6}$

- Coefficients: $3/5$
- $k$: $k^1 / k^5 = k^{-4}$
- $r$: $r^3 / r^6 = r^{-3}$

So:
$$
\frac{3kr^3}{5k^5 r^6} = \frac{3}{5} \cdot k^{-4} \cdot r^{-3} = \frac{3}{5k^4 r^3}
$$

✔ Correct

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7) $\frac{8w^3}{5w^2}$

- Coefficients: $8/5$
- $w$: $w^3 / w^2 = w^{1}$

So:
$$
\frac{8w^3}{5w^2} = \frac{8w}{5}
$$

✔ Correct

---

8) $\frac{4z^{-6}}{8z^6}$

- Coefficients: $4/8 = 1/2$
- $z$: $z^{-6} / z^6 = z^{-12}$

So:
$$
\frac{4z^{-6}}{8z^6} = \frac{1}{2} \cdot z^{-12} = \frac{1}{2z^{12}}
$$

✔ Correct

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9) $\frac{8n^{-2} c^3}{4n^4 c^{-3}}$

- Coefficients: $8/4 = 2$
- $n$: $n^{-2} / n^4 = n^{-6}$
- $c$: $c^3 / c^{-3} = c^{3 - (-3)} = c^6$

So:
$$
\frac{8n^{-2} c^3}{4n^4 c^{-3}} = 2 \cdot n^{-6} \cdot c^6 = \frac{2c^6}{n^6}
$$

✔ Correct

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10) $\frac{3g}{7g^{-4}}$

- Coefficients: $3/7$
- $g$: $g^1 / g^{-4} = g^{1 - (-4)} = g^5$

So:
$$
\frac{3g}{7g^{-4}} = \frac{3}{7} \cdot g^5 = \frac{3g^5}{7}
$$

✔ Correct

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11) $\frac{3d^3}{7d}$

- Coefficients: $3/7$
- $d$: $d^3 / d^1 = d^{2}$

Wait! The given answer is $\frac{3}{7d^4}$ — this is wrong.

Let’s compute:
$$
\frac{3d^3}{7d} = \frac{3}{7} \cdot d^{3-1} = \frac{3}{7} d^2
$$

But the image says: $\frac{3}{7d^4}$ — which is incorrect.

✘ Incorrect — should be $\boxed{\frac{3d^2}{7}}$

---

12) $\frac{9r^5}{5r}$

- Coefficients: $9/5$
- $r$: $r^5 / r^1 = r^4$

So:
$$
\frac{9r^5}{5r} = \frac{9r^4}{5}
$$

✔ Correct

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## ✔ Final Summary:

All problems are correct except #11.

✘ Problem 11 is wrong.

Correct Answer for #11:
$$
\frac{3d^3}{7d} = \frac{3}{7} d^{3-1} = \frac{3d^2}{7}
$$

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## ✔ Corrected Answers:

| Problem | Correct Answer |
|--------|----------------|
| 1) | $\frac{3w^4}{k^{10}}$ |
| 2) | $\frac{1}{3h^2 d^4}$ |
| 3) | $\frac{7}{9r^5}$ |
| 4) | $7^{10}$ |
| 5) | $\frac{1}{7^6}$ |
| 6) | $\frac{3}{5k^4 r^3}$ |
| 7) | $\frac{8w}{5}$ |
| 8) | $\frac{1}{2z^{12}}$ |
| 9) | $\frac{2c^6}{n^6}$ |
| 10) | $\frac{3g^5}{7}$ |
| 11) | $\frac{3d^2}{7}$ ← corrected |
| 12) | $\frac{9r^4}{5}$ |

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✔ Conclusion: All answers are correct except #11, which should be $\boxed{\frac{3d^2}{7}}$.