Exponents Worksheets with Answer Key - Free Printable

Let's solve each problem on the "Properties of Exponents Worksheet" step by step, using exponent rules. Here are the key exponent rules we'll use:

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. Zero Exponent: $a^0 = 1$ (for $a \neq 0$)
5. Negative Exponent: $a^{-n} = \frac{1}{a^n}$, and $\frac{1}{a^{-n}} = a^n$
6. Power of a Product: $(ab)^n = a^n b^n$
7. Power of a Quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

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Solutions:

#### 1. $(x^4)^2$

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

✔ Answer: $x^8$

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#### 2. $x^{-8}$

Use Negative Exponent Rule:
$x^{-8} = \frac{1}{x^8}$

✔ Answer: $\frac{1}{x^8}$

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#### 3. $(a^2)^0$

Use Zero Exponent Rule:
Any non-zero base to the power 0 is 1.
$(a^2)^0 = 1$

✔ Answer: $1$

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#### 4. $2a^2 \cdot 3b$

Multiply coefficients and keep variables separate:
$2 \cdot 3 = 6$, so $6a^2b$

✔ Answer: $6a^2b$

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#### 5. $(4x^2)^{-4}$

Use Power of a Product and Negative Exponent:
$$
(4x^2)^{-4} = 4^{-4} \cdot (x^2)^{-4} = \frac{1}{4^4} \cdot x^{-8} = \frac{1}{256} \cdot \frac{1}{x^8} = \frac{1}{256x^8}
$$

✔ Answer: $\frac{1}{256x^8}$

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#### 6. $(4a^4)^2$

Apply Power of a Product:
$$
(4a^4)^2 = 4^2 \cdot (a^4)^2 = 16 \cdot a^{8} = 16a^8
$$

✔ Answer: $16a^8$

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#### 7. $(4ab)^{-1}$

Use Negative Exponent and Power of a Product:
$$
(4ab)^{-1} = \frac{1}{4ab}
$$

✔ Answer: $\frac{1}{4ab}$

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#### 8. $(a^2b^{-1})^2$

Apply Power of a Product:
$$
(a^2)^2 \cdot (b^{-1})^2 = a^{4} \cdot b^{-2} = \frac{a^4}{b^2}
$$

✔ Answer: $\frac{a^4}{b^2}$

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#### 9. $(6ab)^2$

Use Power of a Product:
$$
6^2 \cdot a^2 \cdot b^2 = 36a^2b^2
$$

✔ Answer: $36a^2b^2$

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#### 10. $\frac{18a^3}{4a}$

Simplify coefficients and use Quotient of Powers:
$$
\frac{18}{4} = \frac{9}{2}, \quad \frac{a^3}{a} = a^{3-1} = a^2
$$
So: $\frac{9}{2}a^2$

✔ Answer: $\frac{9}{2}a^2$

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#### 11. $\frac{2a^3}{a^2}$

Use Quotient of Powers:
$$
\frac{2a^3}{a^2} = 2a^{3-2} = 2a
$$

✔ Answer: $2a$

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#### 12. $\left(\frac{3a^2b^7}{a}\right)^5$

First simplify inside the parentheses:
$$
\frac{3a^2b^7}{a} = 3a^{2-1}b^7 = 3ab^7
$$
Now raise to the 5th power:
$$
(3ab^7)^5 = 3^5 \cdot a^5 \cdot (b^7)^5 = 243a^5b^{35}
$$

✔ Answer: $243a^5b^{35}$

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#### 13. $\frac{a^{-1}}{a^{-8}}$

Use Quotient of Powers:
$$
a^{-1 - (-8)} = a^{-1 + 8} = a^7
$$

✔ Answer: $a^7$

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#### 14. $\frac{x^5y^4}{xy^3}$

Use Quotient of Powers for each variable:
$$
x^{5-1} \cdot y^{4-3} = x^4y^1 = x^4y
$$

✔ Answer: $x^4y$

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#### 15. $-(9a)^0$

First: $(9a)^0 = 1$, since any non-zero expression to the 0 power is 1.

Then apply the negative sign: $-1$

✔ Answer: $-1$

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#### 16. $\frac{1}{2^{-6}}$

Use Negative Exponent Rule:
$$
\frac{1}{2^{-6}} = 2^6 = 64
$$

✔ Answer: $64$

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#### 17. $a^8 \cdot a^{-7}$

Use Product of Powers:
$$
a^{8 + (-7)} = a^1 = a
$$

✔ Answer: $a$

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#### 18. $(a^2b)^4$

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

✔ Answer: $a^8b^4$

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $x^8$ |
| 2 | $\frac{1}{x^8}$ |
| 3 | $1$ |
| 4 | $6a^2b$ |
| 5 | $\frac{1}{256x^8}$ |
| 6 | $16a^8$ |
| 7 | $\frac{1}{4ab}$ |
| 8 | $\frac{a^4}{b^2}$ |
| 9 | $36a^2b^2$ |
| 10 | $\frac{9}{2}a^2$ |
| 11 | $2a$ |
| 12 | $243a^5b^{35}$ |
| 13 | $a^7$ |
| 14 | $x^4y$ |
| 15 | $-1$ |
| 16 | $64$ |
| 17 | $a$ |
| 18 | $a^8b^4$ |

Let me know if you'd like this in a printable format or with explanations for any specific problem!