Practice simplifying expressions with this free Properties of Exponents worksheet featuring 18 varied algebra problems.

Problem: Simplify the given expressions using the properties of exponents.

We will solve each problem step by step, applying the relevant exponent rules. Here are the key exponent rules we will use:
1. Power of a power: $(x^m)^n = x^{m \cdot n}$
2. Negative exponent: $x^{-n} = \frac{1}{x^n}$
3. Zero exponent: $x^0 = 1$ (for $x \neq 0$)
4. Product of powers: $x^m \cdot x^n = x^{m+n}$
5. Quotient of powers: $\frac{x^m}{x^n} = x^{m-n}$
6. Distributive property: $(xy)^n = x^n y^n$

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Problem 1: $(x^4)^2$

- Apply the power of a power rule: $(x^4)^2 = x^{4 \cdot 2} = x^8$.
- Answer: $\boxed{x^8}$

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Problem 2: $x^{-8}$

- Apply the negative exponent rule: $x^{-8} = \frac{1}{x^8}$.
- Answer: $\boxed{\frac{1}{x^8}}$

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Problem 3: $(a^2)^0$

- Apply the zero exponent rule: $(a^2)^0 = 1$.
- Answer: $\boxed{1}$

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Problem 4: $2a^2 \cdot 3b$

- Multiply the coefficients and keep the variables as they are: $2a^2 \cdot 3b = (2 \cdot 3)a^2b = 6a^2b$.
- Answer: $\boxed{6a^2b}$

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Problem 5: $(4x^2)^{-4}$

- Apply the power of a product rule: $(4x^2)^{-4} = 4^{-4} \cdot (x^2)^{-4}$.
- Simplify each part:
- $4^{-4} = \frac{1}{4^4} = \frac{1}{256}$,
- $(x^2)^{-4} = x^{2 \cdot (-4)} = x^{-8} = \frac{1}{x^8}$.
- Combine the results: $(4x^2)^{-4} = \frac{1}{256} \cdot \frac{1}{x^8} = \frac{1}{256x^8}$.
- Answer: $\boxed{\frac{1}{256x^8}}$

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Problem 6: $(4a^4)^2$

- Apply the power of a product rule: $(4a^4)^2 = 4^2 \cdot (a^4)^2$.
- Simplify each part:
- $4^2 = 16$,
- $(a^4)^2 = a^{4 \cdot 2} = a^8$.
- Combine the results: $(4a^4)^2 = 16a^8$.
- Answer: $\boxed{16a^8}$

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Problem 7: $(4ab)^{-1}$

- Apply the negative exponent rule: $(4ab)^{-1} = \frac{1}{4ab}$.
- Answer: $\boxed{\frac{1}{4ab}}$

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

- Apply the power of a product rule: $(a^2b^{-1})^2 = (a^2)^2 \cdot (b^{-1})^2$.
- Simplify each part:
- $(a^2)^2 = a^{2 \cdot 2} = a^4$,
- $(b^{-1})^2 = b^{-1 \cdot 2} = b^{-2} = \frac{1}{b^2}$.
- Combine the results: $(a^2b^{-1})^2 = a^4 \cdot \frac{1}{b^2} = \frac{a^4}{b^2}$.
- Answer: $\boxed{\frac{a^4}{b^2}}$

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Problem 9: $(6ab)^2$

- Apply the power of a product rule: $(6ab)^2 = 6^2 \cdot a^2 \cdot b^2$.
- Simplify each part:
- $6^2 = 36$,
- $a^2$ and $b^2$ remain as they are.
- Combine the results: $(6ab)^2 = 36a^2b^2$.
- Answer: $\boxed{36a^2b^2}$

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Problem 10: $\frac{18a^3}{4a}$

- Simplify the coefficients and apply the quotient of powers rule for the variable $a$:
- Coefficients: $\frac{18}{4} = \frac{9}{2}$,
- Variables: $\frac{a^3}{a} = a^{3-1} = a^2$.
- Combine the results: $\frac{18a^3}{4a} = \frac{9}{2}a^2$.
- Answer: $\boxed{\frac{9}{2}a^2}$

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Problem 11: $\frac{2a^3}{a^2}$

- Apply the quotient of powers rule for the variable $a$:
- Coefficients: $2$ remains as it is,
- Variables: $\frac{a^3}{a^2} = a^{3-2} = a^1 = a$.
- Combine the results: $\frac{2a^3}{a^2} = 2a$.
- Answer: $\boxed{2a}$

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

- Simplify the fraction inside the parentheses first:
- Coefficients: $3$ remains as it is,
- Variables: $\frac{a^2b^7}{a} = a^{2-1} \cdot b^7 = ab^7$.
- Now raise the simplified expression to the power of 5:
- $\left(\frac{3a^2b^7}{a}\right)^5 = (3ab^7)^5$.
- Apply the power of a product rule: $(3ab^7)^5 = 3^5 \cdot a^5 \cdot (b^7)^5$.
- Simplify each part:
- $3^5 = 243$,
- $a^5$ remains as it is,
- $(b^7)^5 = b^{7 \cdot 5} = b^{35}$.
- Combine the results: $(3ab^7)^5 = 243a^5b^{35}$.
- Answer: $\boxed{243a^5b^{35}}$

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Problem 13: $\frac{a^{-1}}{a^{-8}}$

- Apply the quotient of powers rule: $\frac{a^{-1}}{a^{-8}} = a^{-1 - (-8)} = a^{-1 + 8} = a^7$.
- Answer: $\boxed{a^7}$

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

- Apply the quotient of powers rule for both $x$ and $y$:
- For $x$: $\frac{x^5}{x} = x^{5-1} = x^4$,
- For $y$: $\frac{y^4}{y^3} = y^{4-3} = y^1 = y$.
- Combine the results: $\frac{x^5y^4}{xy^3} = x^4y$.
- Answer: $\boxed{x^4y}$

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Problem 15: $-(9a)^0$

- Apply the zero exponent rule: $(9a)^0 = 1$.
- The negative sign remains: $-(9a)^0 = -1$.
- Answer: $\boxed{-1}$

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Problem 16: $\frac{1}{2^{-6}}$

- Apply the negative exponent rule: $\frac{1}{2^{-6}} = 2^6$.
- Simplify: $2^6 = 64$.
- Answer: $\boxed{64}$

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Problem 17: $a^8 \cdot a^{-7}$

- Apply the product of powers rule: $a^8 \cdot a^{-7} = a^{8 + (-7)} = a^{8-7} = a^1 = a$.
- Answer: $\boxed{a}$

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Problem 18: $(a^2b)^4$

- Apply the power of a product rule: $(a^2b)^4 = (a^2)^4 \cdot b^4$.
- Simplify each part:
- $(a^2)^4 = a^{2 \cdot 4} = a^8$,
- $b^4$ remains as it is.
- Combine the results: $(a^2b)^4 = a^8b^4$.
- Answer: $\boxed{a^8b^4}$

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Final Answers:

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

Boxed Final Answer:
\[
\boxed{
\begin{array}{ccc}
1. & x^8 & \\
2. & \frac{1}{x^8} & \\
3. & 1 & \\
4. & 6a^2b & \\
5. & \frac{1}{256x^8} & \\
6. & 16a^8 & \\
7. & \frac{1}{4ab} & \\
8. & \frac{a^4}{b^2} & \\
9. & 36a^2b^2 & \\
10. & \frac{9}{2}a^2 & \\
11. & 2a & \\
12. & 243a^5b^{35} & \\
13. & a^7 & \\
14. & x^4y & \\
15. & -1 & \\
16. & 64 & \\
17. & a & \\
18. & a^8b^4 &
\end{array}
}
\]