Multiplication Property of Exponents worksheets - Free Printable

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

1) $2m^2 \times 2m^3$
* Multiply the numbers: $2 \times 2 = 4$.
* Multiply the variables by adding exponents: $m^2 \times m^3 = m^{2+3} = m^5$.
* Result: $4m^5$

2) $m^4 \times 2m^{-3}$
* The coefficient is just $2$ (since $1 \times 2 = 2$).
* Add exponents for $m$: $4 + (-3) = 1$.
* Result: $2m^1$ or just $2m$

3) $5^2 \times 5^5$
* Keep the base ($5$) and add the exponents: $2 + 5 = 7$.
* Result: $5^7$

4) $4r^{-3} \times 2r^2$
* Multiply numbers: $4 \times 2 = 8$.
* Add exponents for $r$: $-3 + 2 = -1$.
* Result: $8r^{-1}$ (which can also be written as $\frac{8}{r}$)

5) $4n^4 \times 2n^{-3}$
* Multiply numbers: $4 \times 2 = 8$.
* Add exponents for $n$: $4 + (-3) = 1$.
* Result: $8n^1$ or just $8n$

6) $x^2 \times x \times x^{-4}$
* Remember that $x$ is the same as $x^1$.
* Add all exponents: $2 + 1 + (-4) = -1$.
* Result: $x^{-1}$ (or $\frac{1}{x}$)

7) $2k^4 \times 4k$
* Multiply numbers: $2 \times 4 = 8$.
* Add exponents for $k$: $4 + 1 = 5$ (remember $k$ is $k^1$).
* Result: $8k^5$

8) $4v^3 \times vu^2$
* Coefficient is $4$.
* Combine $v$'s: $v^3 \times v^1 = v^4$.
* Keep $u^2$ as it is.
* Result: $4v^4u^2$

9) $(5^6)^0$
* Any non-zero number raised to the power of $0$ is $1$.
* Result: $1$

10) $2y^2 \times 3x$
* Multiply numbers: $2 \times 3 = 6$.
* Combine variables: There are no like bases to combine, so just write them together.
* Result: $6xy^2$ (standard order is alphabetical)

11) $4(xy)^{-1}$
* Apply the exponent $-1$ to everything inside the parenthesis.
* Result: $4x^{-1}y^{-1}$ (or $\frac{4}{xy}$)

12) $(-5x)^2$
* Square the number: $(-5)^2 = 25$ (negative times negative is positive).
* Square the variable: $(x)^2 = x^2$.
* Result: $25x^2$

13) $(3k^4)^4$
* Raise the number to the power: $3^4 = 3 \times 3 \times 3 \times 3 = 81$.
* Multiply the exponents for the variable: $4 \times 4 = 16$.
* Result: $81k^{16}$

14) $(4a^3)^2$
* Square the number: $4^2 = 16$.
* Multiply exponents for $a$: $3 \times 2 = 6$.
* Result: $16a^6$

15) $(a^{-2} \times b^0)^3$
* Simplify inside first: $b^0 = 1$, so it becomes $(a^{-2})^3$.
* Multiply exponents: $-2 \times 3 = -6$.
* Result: $a^{-6}$ (or $\frac{1}{a^6}$)

16) $(4r^0)^4$
* Simplify inside first: $r^0 = 1$, so it becomes $(4 \times 1)^4 = 4^4$.
* Calculate $4^4$: $4 \times 4 \times 4 \times 4 = 256$.
* Result: $256$

17) $(x^2)^0$
* Anything to the power of $0$ is $1$.
* Result: $1$

18) $(2^3)^2$
* Multiply the exponents: $3 \times 2 = 6$. So, $2^6$.
* Calculate $2^6$: $2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$.
* Result: $64$

19) $(2x^2)^{-4}$
* Apply $-4$ to the $2$: $2^{-4} = \frac{1}{2^4} = \frac{1}{16}$.
* Apply $-4$ to $x^2$: $2 \times -4 = -8$. So $x^{-8} = \frac{1}{x^8}$.
* Result: $\frac{1}{16x^8}$

20) $x^2 \times x^6$
* Add exponents: $2 + 6 = 8$.
* Result: $x^8$

21) $-(2)^2$
* Be careful with the order of operations. Exponents happen before the negative sign unless the negative is inside parentheses like $(-2)^2$.
* First calculate $2^2 = 4$.
* Then apply the negative sign.
* Result: $-4$

Final Answer:
1) $4m^5$
2) $2m$
3) $5^7$
4) $8r^{-1}$
5) $8n$
6) $x^{-1}$
7) $8k^5$
8) $4v^4u^2$
9) $1$
10) $6xy^2$
11) $4x^{-1}y^{-1}$
12) $25x^2$
13) $81k^{16}$
14) $16a^6$
15) $a^{-6}$
16) $256$
17) $1$
18) $64$
19) $\frac{1}{16x^8}$
20) $x^8$
21) $-4$