Properties of Exponents Activity {Exponent Rules Worksheet} {Laws of Exponents - Free Printable

Let’s solve each problem one by one using exponent rules. Remember:

- When multiplying same bases: add exponents → $x^a \cdot x^b = x^{a+b}$
- When dividing same bases: subtract exponents → $\frac{x^a}{x^b} = x^{a-b}$
- When raising a power to another power: multiply exponents → $(x^a)^b = x^{a \cdot b}$
- Negative exponents mean reciprocal → $x^{-a} = \frac{1}{x^a}$, and we must eliminate them in final answers.
- Coefficients (numbers) are multiplied normally.

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1. $x^5 \cdot x^2$

Same base, multiply → add exponents:
$x^{5+2} = x^7$

✔ Final: $x^7$

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2. $y^3 \cdot y \cdot y^4$

Remember $y = y^1$. Add all exponents:
$y^{3+1+4} = y^8$

✔ Final: $y^8$

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3. $b^4 \cdot b^{-4}$

Add exponents: $b^{4 + (-4)} = b^0 = 1$

✔ Final: $1$

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4. $7x^3y^2 \cdot 5xy^9$

Multiply coefficients: $7 \cdot 5 = 35$
Add x exponents: $x^{3+1} = x^4$
Add y exponents: $y^{2+9} = y^{11}$

✔ Final: $35x^4y^{11}$

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5. $a^{10} \cdot a^2 \cdot a^{-6}$

Add exponents: $a^{10+2+(-6)} = a^6$

✔ Final: $a^6$

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6. $(z^5)^5$

Power to power → multiply exponents: $z^{5 \cdot 5} = z^{25}$

✔ Final: $z^{25}$

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7. $(b^7)^2$

Multiply exponents: $b^{7 \cdot 2} = b^{14}$

✔ Final: $b^{14}$

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8. $(m^{-8})^{-3}$

Multiply exponents: $m^{(-8) \cdot (-3)} = m^{24}$

✔ Final: $m^{24}$

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9. $(x^2 y^4 m^3)^8$

Apply outer exponent to each inside term:
$x^{2 \cdot 8} = x^{16}$
$y^{4 \cdot 8} = y^{32}$
$m^{3 \cdot 8} = m^{24}$

✔ Final: $x^{16} y^{32} m^{24}$

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10. $(3x^2)^4$

Apply exponent to both coefficient and variable:
$3^4 = 81$
$(x^2)^4 = x^{8}$

✔ Final: $81x^8$

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11. $(2ab)^5$

Apply exponent to each factor:
$2^5 = 32$
$a^5$, $b^5$

✔ Final: $32a^5b^5$

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12. $(2x^3y)^6$

Apply exponent to each:
$2^6 = 64$
$(x^3)^6 = x^{18}$
$y^6$

✔ Final: $64x^{18}y^6$

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13. $(m^7)^4 \cdot m^3$

First: $(m^7)^4 = m^{28}$
Then: $m^{28} \cdot m^3 = m^{31}$

✔ Final: $m^{31}$

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14. $p^2 \cdot (p^5)^2$

First: $(p^5)^2 = p^{10}$
Then: $p^2 \cdot p^{10} = p^{12}$

✔ Final: $p^{12}$

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15. $\frac{x^5}{x^2}$

Subtract exponents: $x^{5-2} = x^3$

✔ Final: $x^3$

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16. $\frac{c^4}{c^8}$

Subtract exponents: $c^{4-8} = c^{-4}$ → but no negative exponents allowed!
So rewrite as: $\frac{1}{c^4}$

✔ Final: $\frac{1}{c^4}$

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17. $\frac{5x^{-4}}{x^{-9}}$

First, handle coefficients: 5 stays.
For x: $x^{-4 - (-9)} = x^{-4+9} = x^5$

✔ Final: $5x^5$

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18. $\frac{x^3 \cdot x^4}{x^2}$

Numerator: $x^{3+4} = x^7$
Then: $\frac{x^7}{x^2} = x^{7-2} = x^5$

✔ Final: $x^5$

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Final Answer:
1. $x^7$
2. $y^8$
3. $1$
4. $35x^4y^{11}$
5. $a^6$
6. $z^{25}$
7. $b^{14}$
8. $m^{24}$
9. $x^{16} y^{32} m^{24}$
10. $81x^8$
11. $32a^5b^5$
12. $64x^{18}y^6$
13. $m^{31}$
14. $p^{12}$
15. $x^3$
16. $\frac{1}{c^4}$
17. $5x^5$
18. $x^5$