Exponent practice worksheet for 8th grade math, focusing on simplifying expressions with positive exponents.

Here are the simplified answers for the exponent practice problems. The goal is to use the rules of exponents to make sure all powers in the final answer are positive.

1) $\frac{2^7}{2^5}$
Subtract the exponents: $7 - 5 = 2$.
Answer: $2^2$ (or 4)

2) $\frac{3}{3^5}$
Think of the top 3 as $3^1$. Subtract exponents: $1 - 5 = -4$. To make it positive, move it to the denominator.
Answer: $\frac{1}{3^4}$ (or $\frac{1}{81}$)

3) $\frac{4^0}{4^4}$
Any number to the power of 0 is 1. So this is $\frac{1}{4^4}$.
Answer: $\frac{1}{4^4}$ (or $\frac{1}{256}$)

4) $\frac{3^0}{3^5}$
Same as above, $3^0 = 1$.
Answer: $\frac{1}{3^5}$ (or $\frac{1}{243}$)

5) $\frac{2^4}{2^0}$
$2^0 = 1$, so this is just $\frac{2^4}{1}$.
Answer: $2^4$ (or 16)

6) $\frac{4a}{ab^5}$
Cancel the $a$ from the top and bottom. The 4 and $b^5$ stay.
Answer: $\frac{4}{b^5}$

7) $\frac{4x^{-1}}{x^3y^0}$
$y^0 = 1$, so ignore it. Move $x^{-1}$ to the bottom to become $x^1$. Combine with $x^3$ ($x^1 \cdot x^3 = x^4$).
Answer: $\frac{4}{x^4}$

8) $\frac{4x^2}{2x^4}$
Divide numbers: $4/2 = 2$. Subtract x exponents: $2 - 4 = -2$. Move $x^2$ to the bottom.
Answer: $\frac{2}{x^2}$

9) $\frac{4m^4}{m^{-3}n^2}$
Move $m^{-3}$ to the top to become $m^3$. Add exponents for m: $m^4 \cdot m^3 = m^7$.
Answer: $\frac{4m^7}{n^2}$

10) $\frac{m^{-7}n^{-3}}{2m^2}$
Move negative exponents to the opposite side. $m^{-7}$ goes down, $n^{-3}$ goes down.
Combine m terms on bottom: $m^7 \cdot m^2 = m^9$.
Answer: $\frac{1}{2m^9n^3}$

11) $\frac{2x^0}{3x^8y^2}$
$x^0 = 1$. Nothing else cancels or combines easily.
Answer: $\frac{2}{3x^8y^2}$

12) $\frac{3x^5y^3}{3x^8y^0}$
Cancel the 3s. $y^0 = 1$. Subtract x exponents: $5 - 8 = -3$. Move $x^3$ to bottom.
Answer: $\frac{y^3}{x^3}$

13) $\frac{3x^4y^{-4}}{4x^3y^4}$
Numbers: $\frac{3}{4}$.
x terms: $x^4 / x^3 = x^1$ (on top).
y terms: $y^{-4} / y^4 = y^{-8}$ (move to bottom as $y^8$).
Answer: $\frac{3x}{4y^8}$

14) $\frac{4x^{-4}}{4y^{-2}}$
Cancel the 4s. Move $x^{-4}$ down ($x^4$). Move $y^{-2}$ up ($y^2$).
Answer: $\frac{y^2}{x^4}$

15) $\frac{4u^0v^{-1}}{4v}$
Cancel 4s. $u^0 = 1$. Combine v terms: $v^{-1} / v^1 = v^{-2}$. Move to bottom.
Answer: $\frac{1}{v^2}$

16) $\frac{2y^4}{2x^1 \cdot 2x^3y^{-4}}$
Simplify bottom first: $2 \cdot 2 = 4$. $x^1 \cdot x^3 = x^4$. Bottom is $4x^4y^{-4}$.
Now divide: $\frac{2y^4}{4x^4y^{-4}}$.
Numbers: $2/4 = 1/2$.
y terms: $y^4 / y^{-4} = y^{4 - (-4)} = y^8$ (on top).
Answer: $\frac{y^8}{2x^4}$

17) $\frac{4u^{-1}v^2}{4u^{-3}v^{-1} \cdot u^{-1}v^4}$
Simplify bottom first:
Numbers: 4.
u terms: $u^{-3} \cdot u^{-1} = u^{-4}$.
v terms: $v^{-1} \cdot v^4 = v^3$.
Bottom is $4u^{-4}v^3$.
Now divide top by bottom: $\frac{4u^{-1}v^2}{4u^{-4}v^3}$.
Cancel 4s.
u terms: $u^{-1} / u^{-4} = u^{-1 - (-4)} = u^3$ (on top).
v terms: $v^2 / v^3 = v^{-1}$ (move to bottom as $v^1$).
Answer: $\frac{u^3}{v}$

18) $\frac{x^{-2}y^{-4} \cdot 3x^5y^2}{4yx^2}$
Simplify top first:
Numbers: 3.
x terms: $x^{-2} \cdot x^5 = x^3$.
y terms: $y^{-4} \cdot y^2 = y^{-2}$.
Top is $3x^3y^{-2}$.
Now divide by bottom $4yx^2$ (which is $4x^2y^1$):
Numbers: $\frac{3}{4}$.
x terms: $x^3 / x^2 = x^1$ (on top).
y terms: $y^{-2} / y^1 = y^{-3}$ (move to bottom as $y^3$).
Answer: $\frac{3x}{4y^3}$

19) $\frac{4a^0b^2}{2a^2b^2 \cdot 2b^{-4}}$
Simplify bottom first:
Numbers: $2 \cdot 2 = 4$.
a terms: $a^2$.
b terms: $b^2 \cdot b^{-4} = b^{-2}$.
Bottom is $4a^2b^{-2}$.
Now divide top by bottom: $\frac{4a^0b^2}{4a^2b^{-2}}$.
Cancel 4s. $a^0 = 1$.
b terms: $b^2 / b^{-2} = b^{2 - (-2)} = b^4$ (on top).
a terms: $1 / a^2$ (stays on bottom).
Answer: $\frac{b^4}{a^2}$

20) $\frac{4x^{-1}y^0}{2x^3y^{-4} \cdot x^5y^{-2}}$
Simplify bottom first:
Numbers: 2.
x terms: $x^3 \cdot x^5 = x^8$.
y terms: $y^{-4} \cdot y^{-2} = y^{-6}$.
Bottom is $2x^8y^{-6}$.
Now divide top by bottom: $\frac{4x^{-1}y^0}{2x^8y^{-6}}$.
Numbers: $4/2 = 2$.
$y^0 = 1$.
x terms: $x^{-1} / x^8 = x^{-9}$ (move to bottom as $x^9$).
y terms: $1 / y^{-6} = y^6$ (move to top).
Answer: $\frac{2y^6}{x^9}$

Final Answer:
1) $2^2$
2) $\frac{1}{3^4}$
3) $\frac{1}{4^4}$
4) $\frac{1}{3^5}$
5) $2^4$
6) $\frac{4}{b^5}$
7) $\frac{4}{x^4}$
8) $\frac{2}{x^2}$
9) $\frac{4m^7}{n^2}$
10) $\frac{1}{2m^9n^3}$
11) $\frac{2}{3x^8y^2}$
12) $\frac{y^3}{x^3}$
13) $\frac{3x}{4y^8}$
14) $\frac{y^2}{x^4}$
15) $\frac{1}{v^2}$
16) $\frac{y^8}{2x^4}$
17) $\frac{u^3}{v}$
18) $\frac{3x}{4y^3}$
19) $\frac{b^4}{a^2}$
20) $\frac{2y^6}{x^9}$