Simplify expressions using exponent rules in this Properties of Exponents Worksheet.

Let's solve each problem on the "Properties of Exponents Worksheet" step by step, using the rules of exponents.

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🔷 Rules of Exponents Used:

1. $(a^m)^n = a^{m \cdot n}$
2. $a^m \cdot a^n = a^{m+n}$
3. $\frac{a^m}{a^n} = a^{m-n}$
4. $a^0 = 1$ (for $a \neq 0$)
5. $a^{-n} = \frac{1}{a^n}$
6. $(ab)^n = a^n b^n$
7. $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

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Now let's simplify each expression:

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1. $(x^4)^2$

Use rule: $(a^m)^n = a^{m \cdot n}$

$$
(x^4)^2 = x^{4 \cdot 2} = x^8
$$

✔ Answer: $x^8$

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2. $x^{-8}$

Negative exponent: $x^{-8} = \frac{1}{x^8}$

✔ Answer: $\frac{1}{x^8}$

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3. $(a^2)^0$

Any nonzero number to the power 0 is 1:

$$
(a^2)^0 = 1
$$

✔ Answer: $1$

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

Multiply coefficients and variables:

$$
(2 \cdot 3)(a^2 \cdot b) = 6a^2b
$$

✔ Answer: $6a^2b$

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

Apply negative exponent and distribute:

$$
= \frac{1}{(4x^2)^4} = \frac{1}{4^4 \cdot (x^2)^4} = \frac{1}{256 \cdot x^8} = \frac{1}{256x^8}
$$

✔ Answer: $\frac{1}{256x^8}$

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

Distribute the exponent:

$$
= 4^2 \cdot (a^4)^2 = 16 \cdot a^{8} = 16a^8
$$

✔ Answer: $16a^8$

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

Negative exponent means reciprocal:

$$
= \frac{1}{4ab}
$$

✔ Answer: $\frac{1}{4ab}$

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

Distribute exponent:

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

✔ Answer: $\frac{a^4}{b^2}$

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9. $(6ab)^2$

Distribute exponent:

$$
= 6^2 \cdot a^2 \cdot b^2 = 36a^2b^2
$$

✔ Answer: $36a^2b^2$

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

Simplify coefficient and subtract exponents:

$$
= \frac{18}{4} \cdot a^{3-1} = \frac{9}{2}a^2
$$

✔ Answer: $\frac{9}{2}a^2$

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

Subtract exponents:

$$
= 2a^{3-2} = 2a
$$

✔ 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 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 rule: $\frac{a^m}{a^n} = a^{m-n}$

$$
= a^{-1 - (-8)} = a^{-1+8} = a^7
$$

✔ Answer: $a^7$

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

Subtract exponents:

$$
= x^{5-1} \cdot y^{4-3} = x^4y
$$

✔ Answer: $x^4y$

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15. $-(9a)^0$

Note: The negative sign is outside the parentheses.

$$
(9a)^0 = 1 \quad \Rightarrow \quad -(9a)^0 = -1
$$

✔ Answer: $-1$

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

Recall: $a^{-n} = \frac{1}{a^n}$, so:

$$
\frac{1}{2^{-6}} = 2^6 = 64
$$

✔ Answer: $64$

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

Add exponents:

$$
= a^{8 + (-7)} = a^1 = a
$$

✔ Answer: $a$

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

Distribute exponent:

$$
= (a^2)^4 \cdot b^4 = a^{8}b^4
$$

✔ 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 | $a^8b^4$ |

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