Exponents Worksheets - Free Printable

Let's solve each of these exponent problems step by step using the laws of exponents. We'll simplify each expression and write the answers with positive exponents only.

---

Exponent Rules Recap:

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. $ (ab)^n = a^n b^n $
5. $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $
6. $ a^{-n} = \frac{1}{a^n} $
7. $ \frac{1}{a^{-n}} = a^n $

We’ll apply these rules to each problem.

---

1) $ \left( \frac{z^2 y^4}{x^3 y^2} \right)^3 $

Simplify inside first:

$$
\frac{z^2 y^4}{x^3 y^2} = \frac{z^2 y^{4-2}}{x^3} = \frac{z^2 y^2}{x^3}
$$

Now raise to the 3rd power:

$$
\left( \frac{z^2 y^2}{x^3} \right)^3 = \frac{(z^2)^3 (y^2)^3}{(x^3)^3} = \frac{z^6 y^6}{x^9}
$$

✔ Answer: $ \boxed{\frac{z^6 y^6}{x^9}} $

---

2) $ (a^2 b^3)(ab^2)^3 $

First, expand $ (ab^2)^3 $:

$$
(ab^2)^3 = a^3 (b^2)^3 = a^3 b^6
$$

Now multiply:

$$
(a^2 b^3)(a^3 b^6) = a^{2+3} b^{3+6} = a^5 b^9
$$

✔ Answer: $ \boxed{a^5 b^9} $

---

3) $ \left( \frac{8m^4 n^{-2}}{2mn^3} \right)^{-1} $

Simplify inside first:

$$
\frac{8m^4 n^{-2}}{2mn^3} = \frac{8}{2} \cdot \frac{m^4}{m} \cdot \frac{n^{-2}}{n^3} = 4 m^{3} n^{-5}
$$

Now apply the exponent $-1$:

$$
(4 m^3 n^{-5})^{-1} = 4^{-1} m^{-3} n^{5} = \frac{1}{4} m^{-3} n^5
$$

Convert to positive exponents:

$$
= \frac{n^5}{4m^3}
$$

✔ Answer: $ \boxed{\frac{n^5}{4m^3}} $

---

4) $ (5p^2 q^3)(2p^4 q)^2 $

First, square the second term:

$$
(2p^4 q)^2 = 2^2 (p^4)^2 q^2 = 4 p^8 q^2
$$

Now multiply:

$$
(5p^2 q^3)(4 p^8 q^2) = 5 \cdot 4 \cdot p^{2+8} \cdot q^{3+2} = 20 p^{10} q^5
$$

✔ Answer: $ \boxed{20p^{10}q^5} $

---

5) $ \frac{(2a^{-1})(2b^2)}{4b^{-1}} $

Numerator: $ 2a^{-1} \cdot 2b^2 = 4 a^{-1} b^2 $

Denominator: $ 4b^{-1} $

So:

$$
\frac{4 a^{-1} b^2}{4 b^{-1}} = \frac{4}{4} \cdot a^{-1} \cdot \frac{b^2}{b^{-1}} = 1 \cdot a^{-1} \cdot b^{2 - (-1)} = a^{-1} b^3
$$

Convert to positive exponents:

$$
= \frac{b^3}{a}
$$

✔ Answer: $ \boxed{\frac{b^3}{a}} $

---

6) $ (a^{-4})(a^{-3})(a^{-2})(a^{-1}) $

Add exponents:

$$
a^{-4 + (-3) + (-2) + (-1)} = a^{-10}
$$

Convert to positive exponent:

$$
= \frac{1}{a^{10}}
$$

✔ Answer: $ \boxed{\frac{1}{a^{10}}} $

---

7) $ \left( \frac{40 x^2}{31 y^4} \right)^{-5} $

Apply negative exponent:

$$
= \left( \frac{31 y^4}{40 x^2} \right)^5 = \frac{(31)^5 y^{20}}{(40)^5 x^{10}}
$$

Leave as is or write in simplified form:

$$
= \frac{31^5 y^{20}}{40^5 x^{10}}
$$

(We don’t need to compute the actual numbers.)

✔ Answer: $ \boxed{\frac{31^5 y^{20}}{40^5 x^{10}}} $

---

8) $ \left( \frac{z^{-2} a^4}{x^3 z^4} \right)(3x^{-2}) $

First simplify the fraction:

$$
\frac{z^{-2} a^4}{x^3 z^4} = a^4 \cdot z^{-2 - 4} \cdot x^{-3} = a^4 z^{-6} x^{-3}
$$

Now multiply by $ 3x^{-2} $:

$$
3x^{-2} \cdot a^4 z^{-6} x^{-3} = 3 a^4 z^{-6} x^{-2 - 3} = 3 a^4 z^{-6} x^{-5}
$$

Convert to positive exponents:

$$
= \frac{3a^4}{x^5 z^6}
$$

✔ Answer: $ \boxed{\frac{3a^4}{x^5 z^6}} $

---

9) $ (a^{-2} a^3) \left( \frac{2m^{-3} a^{-1}}{m^2 a^{-4}} \right) $

First, simplify $ a^{-2} a^3 = a^{1} $

Now simplify the fraction:

$$
\frac{2m^{-3} a^{-1}}{m^2 a^{-4}} = 2 \cdot m^{-3 - 2} \cdot a^{-1 - (-4)} = 2 m^{-5} a^{3}
$$

Now multiply:

$$
a \cdot 2 m^{-5} a^3 = 2 a^{1+3} m^{-5} = 2 a^4 m^{-5}
$$

Convert:

$$
= \frac{2a^4}{m^5}
$$

✔ Answer: $ \boxed{\frac{2a^4}{m^5}} $

---

10) $ \frac{8p^3 q^{-2} r^3}{(2p^{-1} q^{-1} r^4)^2} $

First, square the denominator:

$$
(2p^{-1} q^{-1} r^4)^2 = 2^2 (p^{-1})^2 (q^{-1})^2 (r^4)^2 = 4 p^{-2} q^{-2} r^8
$$

Now divide:

$$
\frac{8p^3 q^{-2} r^3}{4 p^{-2} q^{-2} r^8} = \frac{8}{4} \cdot p^{3 - (-2)} \cdot q^{-2 - (-2)} \cdot r^{3 - 8} = 2 \cdot p^5 \cdot q^0 \cdot r^{-5}
$$

Since $ q^0 = 1 $, we have:

$$
2 p^5 r^{-5} = \frac{2p^5}{r^5}
$$

✔ Answer: $ \boxed{\frac{2p^5}{r^5}} $

---

11) $ \left( \frac{3x^{-1} y^2}{4x^2 y^{-3}} \right)^{-3} $

Simplify inside first:

$$
\frac{3x^{-1} y^2}{4x^2 y^{-3}} = \frac{3}{4} \cdot x^{-1 - 2} \cdot y^{2 - (-3)} = \frac{3}{4} x^{-3} y^5
$$

Now apply exponent $-3$:

$$
\left( \frac{3}{4} x^{-3} y^5 \right)^{-3} = \left( \frac{3}{4} \right)^{-3} \cdot (x^{-3})^{-3} \cdot (y^5)^{-3} = \left( \frac{4}{3} \right)^3 \cdot x^9 \cdot y^{-15}
$$

Compute $ \left( \frac{4}{3} \right)^3 = \frac{64}{27} $

So:

$$
= \frac{64}{27} x^9 y^{-15} = \frac{64 x^9}{27 y^{15}}
$$

✔ Answer: $ \boxed{\frac{64x^9}{27y^{15}}} $

---

12) $ (3r^2 s^3)^2 (2rs)^{-2} (1s^4)^{-3} $

Break it down:

1. $ (3r^2 s^3)^2 = 3^2 (r^2)^2 (s^3)^2 = 9 r^4 s^6 $
2. $ (2rs)^{-2} = 2^{-2} r^{-2} s^{-2} = \frac{1}{4} r^{-2} s^{-2} $
3. $ (1s^4)^{-3} = (s^4)^{-3} = s^{-12} $

Now multiply all:

$$
9 r^4 s^6 \cdot \frac{1}{4} r^{-2} s^{-2} \cdot s^{-12} = \frac{9}{4} r^{4-2} s^{6 - 2 - 12} = \frac{9}{4} r^2 s^{-8}
$$

Convert:

$$
= \frac{9r^2}{4s^8}
$$

✔ Answer: $ \boxed{\frac{9r^2}{4s^8}} $

---

13) $ (4a^2)^{-1}(a^{-3})^{-1}(a^{-4})^2 $

Simplify each term:

1. $ (4a^2)^{-1} = 4^{-1} a^{-2} = \frac{1}{4} a^{-2} $
2. $ (a^{-3})^{-1} = a^{3} $
3. $ (a^{-4})^2 = a^{-8} $

Now multiply:

$$
\frac{1}{4} a^{-2} \cdot a^3 \cdot a^{-8} = \frac{1}{4} a^{-2 + 3 - 8} = \frac{1}{4} a^{-7} = \frac{1}{4a^7}
$$

✔ Answer: $ \boxed{\frac{1}{4a^7}} $

---

14) $ \left( \frac{6a^{-3} b^4}{2xy^{-3}} \right)^{-2} $

Simplify inside:

$$
\frac{6a^{-3} b^4}{2xy^{-3}} = 3 a^{-3} b^4 x^{-1} y^{3}
$$

Now apply exponent $-2$:

$$
(3 a^{-3} b^4 x^{-1} y^3)^{-2} = 3^{-2} a^{6} b^{-8} x^{2} y^{-6} = \frac{1}{9} a^6 b^{-8} x^2 y^{-6}
$$

Convert to positive exponents:

$$
= \frac{a^6 x^2}{9 b^8 y^6}
$$

✔ Answer: $ \boxed{\frac{a^6 x^2}{9b^8 y^6}} $

---

15) $ \frac{(2a^{-2} b)(4ab^{-3})}{4c^{-1}} $

Numerator: $ (2a^{-2} b)(4ab^{-3}) = 8 a^{-2+1} b^{1-3} = 8 a^{-1} b^{-2} $

Denominator: $ 4c^{-1} $

So:

$$
\frac{8 a^{-1} b^{-2}}{4 c^{-1}} = 2 a^{-1} b^{-2} c^{1} = \frac{2c}{a b^2}
$$

✔ Answer: $ \boxed{\frac{2c}{ab^2}} $

---

✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ \frac{z^6 y^6}{x^9} $ |
| 2 | $ a^5 b^9 $ |
| 3 | $ \frac{n^5}{4m^3} $ |
| 4 | $ 20p^{10}q^5 $ |
| 5 | $ \frac{b^3}{a} $ |
| 6 | $ \frac{1}{a^{10}} $ |
| 7 | $ \frac{31^5 y^{20}}{40^5 x^{10}} $ |
| 8 | $ \frac{3a^4}{x^5 z^6} $ |
| 9 | $ \frac{2a^4}{m^5} $ |
| 10 | $ \frac{2p^5}{r^5} $ |
| 11 | $ \frac{64x^9}{27y^{15}} $ |
| 12 | $ \frac{9r^2}{4s^8} $ |
| 13 | $ \frac{1}{4a^7} $ |
| 14 | $ \frac{a^6 x^2}{9b^8 y^6} $ |
| 15 | $ \frac{2c}{ab^2} $ |

Let me know if you'd like these formatted for printing or want explanations for any specific one!