Power of Quotients Worksheets - 15 Worksheets Library - Free Printable

Let's solve each problem in the "Power of Quotient" worksheet. The key rule we'll use is:

> Power of a Quotient Rule:
> $$
> \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}
> $$

Additionally, we’ll apply exponent rules:
- $ (ab)^n = a^n b^n $
- $ (a^m)^n = a^{m \cdot n} $
- Negative exponents: $ a^{-n} = \frac{1}{a^n} $

We’ll simplify each expression step by step.

---

1) $ \left( \frac{2x^3}{3y} \right)^2 $

$$
= \frac{(2x^3)^2}{(3y)^2} = \frac{2^2 \cdot (x^3)^2}{3^2 \cdot y^2} = \frac{4x^6}{9y^2}
$$

✔ Answer: $ \frac{4x^6}{9y^2} $

---

2) $ \left( \frac{4a^2b}{5c} \right)^3 $

$$
= \frac{(4a^2b)^3}{(5c)^3} = \frac{4^3 \cdot (a^2)^3 \cdot b^3}{5^3 \cdot c^3} = \frac{64a^6b^3}{125c^3}
$$

✔ Answer: $ \frac{64a^6b^3}{125c^3} $

---

3) $ \left( \frac{3p^2}{5c} \right)^4 $

$$
= \frac{(3p^2)^4}{(5c)^4} = \frac{3^4 \cdot (p^2)^4}{5^4 \cdot c^4} = \frac{81p^8}{625c^4}
$$

✔ Answer: $ \frac{81p^8}{625c^4} $

---

4) $ \left( \frac{5m^2n^2}{2p} \right)^2 $

$$
= \frac{(5m^2n^2)^2}{(2p)^2} = \frac{5^2 \cdot (m^2)^2 \cdot (n^2)^2}{2^2 \cdot p^2} = \frac{25m^4n^4}{4p^2}
$$

✔ Answer: $ \frac{25m^4n^4}{4p^2} $

---

5) $ \left( \frac{2x}{y^3} \right)^3 $

$$
= \frac{(2x)^3}{(y^3)^3} = \frac{8x^3}{y^9}
$$

✔ Answer: $ \frac{8x^3}{y^9} $

---

6) $ \left( \frac{ab^2}{3cd} \right)^{-2} $

Negative exponent → flip and make positive:

$$
= \left( \frac{3cd}{ab^2} \right)^2 = \frac{(3cd)^2}{(ab^2)^2} = \frac{9c^2d^2}{a^2b^4}
$$

✔ Answer: $ \frac{9c^2d^2}{a^2b^4} $

---

7) $ \left( \frac{-3pq}{2r^2} \right)^4 $

Note: $ (-3pq)^4 = (+) $ since even power.

$$
= \frac{(-3pq)^4}{(2r^2)^4} = \frac{81p^4q^4}{16r^8}
$$

✔ Answer: $ \frac{81p^4q^4}{16r^8} $

---

8) $ \left( \frac{2xy^2}{3z} \right)^3 $

$$
= \frac{(2xy^2)^3}{(3z)^3} = \frac{8x^3y^6}{27z^3}
$$

✔ Answer: $ \frac{8x^3y^6}{27z^3} $

---

9) $ \left( \frac{5a^3}{b^{-2}} \right)^2 $

First simplify inside: $ b^{-2} = \frac{1}{b^2} $, so:

$$
\frac{5a^3}{b^{-2}} = 5a^3 \cdot b^2
$$

Now square it:

$$
(5a^3 b^2)^2 = 25a^6b^4
$$

✔ Answer: $ 25a^6b^4 $

---

10) $ \left( \frac{3uv}{2w^4} \right)^3 $

$$
= \frac{(3uv)^3}{(2w^4)^3} = \frac{27u^3v^3}{8w^{12}}
$$

✔ Answer: $ \frac{27u^3v^3}{8w^{12}} $

---

11) $ \left( \frac{-4x^2}{y} \right)^2 $

Even power removes negative sign:

$$
= \frac{(-4x^2)^2}{y^2} = \frac{16x^4}{y^2}
$$

✔ Answer: $ \frac{16x^4}{y^2} $

---

12) $ \left( \frac{2m}{3p^2} \right)^3 $

$$
= \frac{(2m)^3}{(3p^2)^3} = \frac{8m^3}{27p^6}
$$

✔ Answer: $ \frac{8m^3}{27p^6} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ \frac{4x^6}{9y^2} $ |
| 2 | $ \frac{64a^6b^3}{125c^3} $ |
| 3 | $ \frac{81p^8}{625c^4} $ |
| 4 | $ \frac{25m^4n^4}{4p^2} $ |
| 5 | $ \frac{8x^3}{y^9} $ |
| 6 | $ \frac{9c^2d^2}{a^2b^4} $ |
| 7 | $ \frac{81p^4q^4}{16r^8} $ |
| 8 | $ \frac{8x^3y^6}{27z^3} $ |
| 9 | $ 25a^6b^4 $ |
| 10 | $ \frac{27u^3v^3}{8w^{12}} $ |
| 11 | $ \frac{16x^4}{y^2} $ |
| 12 | $ \frac{8m^3}{27p^6} $ |

Let me know if you'd like this formatted as a printable answer sheet!