Exponents Worksheets with Answer Key - Free Printable

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

1) $4r \cdot 5r^2$
* Multiply the numbers: $4 \cdot 5 = 20$.
* Multiply the variables: $r \cdot r^2 = r^{1+2} = r^3$.
* Answer: $20r^3$

2) $2p^4 \cdot 4p^4$
* Multiply the numbers: $2 \cdot 4 = 8$.
* Multiply the variables: $p^4 \cdot p^4 = p^{4+4} = p^8$.
* Answer: $8p^8$

3) $10xy^3 \cdot 8x^5y^3$
* Multiply the numbers: $10 \cdot 8 = 80$.
* Multiply the $x$'s: $x^1 \cdot x^5 = x^{1+5} = x^6$.
* Multiply the $y$'s: $y^3 \cdot y^3 = y^{3+3} = y^6$.
* Answer: $80x^6y^6$

4) $7x^{-4}y^4 \cdot 5y^{-2}$
* Multiply the numbers: $7 \cdot 5 = 35$.
* The $x$ term stays as $x^{-4}$.
* Multiply the $y$'s: $y^4 \cdot y^{-2} = y^{4-2} = y^2$.
* Current result: $35x^{-4}y^2$.
* Make exponents positive: Move $x^{-4}$ to the bottom of a fraction to make it $x^4$.
* Answer: $\frac{35y^2}{x^4}$

5) $\frac{x^0y^{-3}}{z^{-3}}$
* Simplify $x^0$: Anything to the power of 0 is 1, so this disappears.
* Handle negative exponents:
* $y^{-3}$ moves from top to bottom to become $y^3$.
* $z^{-3}$ moves from bottom to top to become $z^3$.
* Answer: $\frac{z^3}{y^3}$

6) $\frac{2x^{-3}y^2}{4x^{-4}y^{-1}}$
* Simplify numbers: $\frac{2}{4}$ reduces to $\frac{1}{2}$.
* Simplify $x$'s: $\frac{x^{-3}}{x^{-4}} = x^{-3 - (-4)} = x^{-3+4} = x^1 = x$. (The $x$ goes on top).
* Simplify $y$'s: $\frac{y^2}{y^{-1}} = y^{2 - (-1)} = y^{2+1} = y^3$. (The $y$ goes on top).
* Combine them over the denominator 2.
* Answer: $\frac{xy^3}{2}$

7) $7p^3 \cdot 10q^3p^5 \cdot 8qp^3$
* Multiply the numbers: $7 \cdot 10 \cdot 8 = 560$.
* Multiply the $p$'s: $p^3 \cdot p^5 \cdot p^3 = p^{3+5+3} = p^{11}$.
* Multiply the $q$'s: $q^3 \cdot q^1 = q^{3+1} = q^4$.
* Answer: $560p^{11}q^4$

8) $(7x^2)^{-6}$
* Apply the exponent to everything inside the parenthesis.
* Number part: $7^{-6}$. To make it positive, move it to the bottom: $\frac{1}{7^6}$.
* Variable part: $(x^2)^{-6} = x^{2 \cdot -6} = x^{-12}$. Move to bottom: $\frac{1}{x^{12}}$.
* Answer: $\frac{1}{7^6x^{12}}$

9) $x^2y^{-4} \cdot x^3y^2$
* Multiply $x$'s: $x^2 \cdot x^3 = x^{2+3} = x^5$.
* Multiply $y$'s: $y^{-4} \cdot y^2 = y^{-4+2} = y^{-2}$.
* Make exponent positive: Move $y^{-2}$ to the bottom to become $y^2$.
* Answer: $\frac{x^5}{y^2}$

10) $(2a^4b^{-3})^{-1}$
* Apply the $-1$ exponent to every part inside. This essentially flips the fraction.
* Number: $2^{-1}$ becomes $\frac{1}{2}$.
* $a$ term: $(a^4)^{-1} = a^{-4}$ becomes $\frac{1}{a^4}$.
* $b$ term: $(b^{-3})^{-1} = b^3$ stays on top.
* Answer: $\frac{b^3}{2a^4}$

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Final Answer:
1) $20r^3$
2) $8p^8$
3) $80x^6y^6$
4) $\frac{35y^2}{x^4}$
5) $\frac{z^3}{y^3}$
6) $\frac{xy^3}{2}$
7) $560p^{11}q^4$
8) $\frac{1}{7^6x^{12}}$
9) $\frac{x^5}{y^2}$
10) $\frac{b^3}{2a^4}$