Exponents Worksheets with Answer Key - Free Printable

Here are the step-by-step solutions for each problem on the worksheet.

1. $(a^4)^2$
* Rule: When you have a power raised to another power, you multiply the exponents: $(x^m)^n = x^{m \cdot n}$.
* Step: Multiply $4 \times 2$.
* Result: $a^8$

2. $p^6 \cdot p^{14}$
* Rule: When multiplying terms with the same base, you add the exponents: $x^m \cdot x^n = x^{m + n}$.
* Step: Add $6 + 14$.
* Result: $p^{20}$

3. $\frac{p^7}{p^5}$
* Rule: When dividing terms with the same base, you subtract the bottom exponent from the top exponent: $\frac{x^m}{x^n} = x^{m - n}$.
* Step: Subtract $7 - 5$.
* Result: $p^2$

4. $(z^2)^3$
* Rule: Power of a power rule (multiply exponents).
* Step: Multiply $2 \times 3$.
* Result: $z^6$

5. $\frac{q^{10}}{q^6}$
* Rule: Quotient rule (subtract exponents).
* Step: Subtract $10 - 6$.
* Result: $q^4$

6. $\frac{l^2}{l}$
* Rule: Quotient rule. Remember that $l$ is the same as $l^1$.
* Step: Subtract $2 - 1$.
* Result: $l$ (or $l^1$)

7. $(x^3b)^4(xb^6)^2$
* Step 1: Distribute the outer exponents to everything inside the parentheses using the power rule $(xy)^n = x^n y^n$.
* First part: $(x^3)^4 \cdot b^4 = x^{12}b^4$
* Second part: $x^2 \cdot (b^6)^2 = x^2b^{12}$
* Step 2: Multiply the results together by adding exponents for like bases.
* $x$: $12 + 2 = 14$
* $b$: $4 + 12 = 16$
* Result: $x^{14}b^{16}$

8. $(\frac{a^2b}{b^{-3}c^4})^3 (a^{-3}b)^{-2}$
* Step 1: Simplify the first part inside the parenthesis before applying the cube. Combine the $b$ terms ($\frac{b^1}{b^{-3}} = b^{1 - (-3)} = b^4$).
* Inside becomes: $\frac{a^2 b^4}{c^4}$
* Apply the cube: $(\frac{a^2 b^4}{c^4})^3 = \frac{a^6 b^{12}}{c^{12}}$
* Step 2: Simplify the second part.
* $(a^{-3}b)^{-2} = a^{(-3)(-2)} b^{(1)(-2)} = a^6 b^{-2} = \frac{a^6}{b^2}$
* Step 3: Multiply the two simplified parts.
* $\frac{a^6 b^{12}}{c^{12}} \cdot \frac{a^6}{b^2}$
* Combine $a$: $a^6 \cdot a^6 = a^{12}$
* Combine $b$: $\frac{b^{12}}{b^2} = b^{10}$
* $c$ stays in the denominator.
* Result: $\frac{a^{12}b^{10}}{c^{12}}$

9. $(\frac{x^2y^7}{x^{-2}y^4})^2$
* Step 1: Simplify inside the parentheses first.
* $x$: $\frac{x^2}{x^{-2}} = x^{2 - (-2)} = x^4$
* $y$: $\frac{y^7}{y^4} = y^{7 - 4} = y^3$
* Inside becomes: $x^4 y^3$
* Step 2: Apply the square outside.
* $(x^4 y^3)^2 = x^{4 \cdot 2} y^{3 \cdot 2}$
* Result: $x^8 y^6$

10. $(\frac{p^3q^5}{r^7}) \cdot (\frac{p^2r^0q^3}{p^4r^2})^3$
* Step 1: Simplify the term with the exponent of 3 first. Note that $r^0 = 1$.
* Inside: $\frac{p^2 q^3}{p^4 r^2}$
* Apply the cube: $\frac{p^{2 \cdot 3} q^{3 \cdot 3}}{p^{4 \cdot 3} r^{2 \cdot 3}} = \frac{p^6 q^9}{p^{12} r^6}$
* Simplify the $p$'s inside this fraction: $\frac{q^9}{p^6 r^6}$ (since $p^6 / p^{12} = 1/p^6$)
* Step 2: Multiply this result by the first term $(\frac{p^3q^5}{r^7})$.
* $\frac{p^3 q^5}{r^7} \cdot \frac{q^9}{p^6 r^6}$
* Step 3: Combine like bases.
* $p$: $\frac{p^3}{p^6} = \frac{1}{p^3}$
* $q$: $q^5 \cdot q^9 = q^{14}$
* $r$: $r^7 \cdot r^6 = r^{13}$ (in the denominator)
* Result: $\frac{q^{14}}{p^3 r^{13}}$

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Final Answer:
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) $x^8y^6$
10) $\frac{q^{14}}{p^3r^{13}}$