Exponents Worksheets with Answer Key - Free Printable

Let's solve each problem step by step using the laws of exponents. We'll simplify each expression and write the final answer with positive exponents only.

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🔷 Laws of Exponents Used:

1. Power of a Power: $(a^m)^n = a^{m \cdot n}$
2. Product of Powers: $a^m \cdot a^n = a^{m+n}$
3. Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$
4. Power of a Product: $(ab)^n = a^n b^n$
5. Power of a Quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
6. Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$, so move to denominator or numerator accordingly.
7. Zero Exponent: $a^0 = 1$ (if $a \neq 0$)

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✔ Problem ①: $(a^4)^2$

Use Power of a Power:
$$
(a^4)^2 = a^{4 \cdot 2} = a^8
$$

✔ Answer: $a^8$

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✔ Problem ②: $p^6 \cdot p^{14}$

Use Product of Powers:
$$
p^6 \cdot p^{14} = p^{6+14} = p^{20}
$$

✔ Answer: $p^{20}$

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✔ Problem ③: $\frac{p^7}{p^5}$

Use Quotient of Powers:
$$
\frac{p^7}{p^5} = p^{7-5} = p^2
$$

✔ Answer: $p^2$

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✔ Problem ④: $(z^2)^3$

Use Power of a Power:
$$
(z^2)^3 = z^{2 \cdot 3} = z^6
$$

✔ Answer: $z^6$

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✔ Problem ⑤: $\frac{q^{10}}{q^6}$

Use Quotient of Powers:
$$
\frac{q^{10}}{q^6} = q^{10-6} = q^4
$$

✔ Answer: $q^4$

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✔ Problem ⑥: $\frac{l^2}{l}$

Note: $l = l^1$, so:
$$
\frac{l^2}{l^1} = l^{2-1} = l^1 = l
$$

✔ Answer: $l$

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✔ Problem ⑦: $(x^3 b)^4 (x b^6)^2$

First, apply Power of a Product to each term:

- $(x^3 b)^4 = (x^3)^4 \cdot b^4 = x^{12} b^4$
- $(x b^6)^2 = x^2 \cdot (b^6)^2 = x^2 b^{12}$

Now multiply:
$$
x^{12} b^4 \cdot x^2 b^{12} = x^{12+2} b^{4+12} = x^{14} b^{16}
$$

✔ Answer: $x^{14} b^{16}$

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✔ Problem ⑧: $\left(\frac{a^2 b}{b^{-3} c^4}\right)^3 (a^{-3} b)^{-2}$

We’ll simplify step by step.

#### Step 1: Simplify inside the first parentheses:
$$
\frac{a^2 b}{b^{-3} c^4}
$$

Simplify the $b$ terms: $b / b^{-3} = b^{1 - (-3)} = b^{1+3} = b^4$

So:
$$
\frac{a^2 b}{b^{-3} c^4} = \frac{a^2 b^4}{c^4}
$$

Now raise to the 3rd power:
$$
\left(\frac{a^2 b^4}{c^4}\right)^3 = \frac{(a^2)^3 (b^4)^3}{(c^4)^3} = \frac{a^6 b^{12}}{c^{12}}
$$

#### Step 2: Simplify $(a^{-3} b)^{-2}$

Apply power to both parts:
$$
(a^{-3})^{-2} \cdot b^{-2} = a^{(-3)(-2)} \cdot b^{-2} = a^6 b^{-2}
$$

Now combine with previous result:
$$
\frac{a^6 b^{12}}{c^{12}} \cdot a^6 b^{-2} = a^{6+6} b^{12 + (-2)} c^{-12} = a^{12} b^{10} c^{-12}
$$

Now convert negative exponent to positive:
$$
c^{-12} = \frac{1}{c^{12}}
$$

So:
$$
\frac{a^{12} b^{10}}{c^{12}}
$$

✔ Answer: $\frac{a^{12} b^{10}}{c^{12}}$

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✔ Problem ⑨: $\left(\frac{x^2 y^7}{x^{-2} y^4}\right)^2$

First simplify inside the parentheses.

Simplify $x^2 / x^{-2} = x^{2 - (-2)} = x^{4}$

Simplify $y^7 / y^4 = y^{7-4} = y^3$

So:
$$
\frac{x^2 y^7}{x^{-2} y^4} = x^4 y^3
$$

Now square it:
$$
(x^4 y^3)^2 = x^{8} y^{6}
$$

✔ Answer: $x^8 y^6$

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✔ Problem ⑩: $\left(\frac{p^3 q^5}{r^7}\right) \cdot \left(\frac{p^2 r^0 q^3}{p^4 r^2}\right)^3$

We’ll simplify each part.

#### First term: $\frac{p^3 q^5}{r^7}$ — leave as is for now.

#### Second term: $\left(\frac{p^2 r^0 q^3}{p^4 r^2}\right)^3$

Note: $r^0 = 1$, so:
$$
\frac{p^2 \cdot 1 \cdot q^3}{p^4 r^2} = \frac{p^2 q^3}{p^4 r^2} = p^{2-4} q^3 r^{-2} = p^{-2} q^3 r^{-2}
$$

Now raise to the 3rd power:
$$
(p^{-2} q^3 r^{-2})^3 = p^{-6} q^9 r^{-6}
$$

Now multiply by the first term:
$$
\frac{p^3 q^5}{r^7} \cdot p^{-6} q^9 r^{-6}
= p^{3 + (-6)} q^{5+9} r^{-7 + (-6)} = p^{-3} q^{14} r^{-13}
$$

Convert negative exponents to positive:
$$
p^{-3} = \frac{1}{p^3}, \quad r^{-13} = \frac{1}{r^{13}}
$$

So:
$$
\frac{q^{14}}{p^3 r^{13}}
$$

✔ Answer: $\frac{q^{14}}{p^3 r^{13}}$

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✔ Final Answers:

1. $a^8$
2. $p^{20}$
3. $p^2$
4. $z^6$
5. $q^4$
6. $l$
7. $x^{14} b^{16}$
8. $\frac{a^{12} b^{10}}{c^{12}}$
9. $x^8 y^6$
10. $\frac{q^{14}}{p^3 r^{13}}$

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