Exponent Division Worksheets - 15 Worksheets Library - Free Printable

To solve these problems, we use the rule for dividing exponents with the same base:

> When you divide two powers with the same base, subtract the exponent in the denominator from the exponent in the numerator:
> a^m ÷ a^n = a^(m - n)

Also, remember:
- If the result is a negative exponent, write it as a fraction: a^(-n) = 1 / a^n
- Any number (except 0) raised to the power of 0 is 1

Let’s go through each problem one by one.

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1) 5⁷ / 5⁹
Same base (5), so subtract exponents: 7 - 9 = -2 → 5^(-2) = 1 / 5² = 1/25

2) 3⁵ / 3¹²
5 - 12 = -7 → 3^(-7) = 1 / 3⁷

3) 2⁸ / 2¹¹
8 - 11 = -3 → 2^(-3) = 1 / 2³ = 1/8

4) (-7)⁴ / (-7)¹²
Same base (-7), so 4 - 12 = -8 → (-7)^(-8) = 1 / (-7)⁸
Note: Since the exponent is even, (-7)⁸ is positive, but we leave it as is unless asked to simplify further.

5) 4³ / 4⁷
3 - 7 = -4 → 4^(-4) = 1 / 4⁴

6) (-3)⁵ / (-3)¹¹
5 - 11 = -6 → (-3)^(-6) = 1 / (-3)⁶
Again, even exponent → positive, but we keep form.

7) 6¹² / 6²⁰
12 - 20 = -8 → 6^(-8) = 1 / 6⁸

8) (-5)⁴ / (-5)⁷
4 - 7 = -3 → (-5)^(-3) = 1 / (-5)³ = 1 / (-125) = -1/125

Wait — let’s check: (-5)³ = -125, so 1 / (-125) = -1/125

But note: sometimes teachers prefer to write with positive exponents only. Let’s see if we can write it as -1/(5³) or just leave as 1/(-5)³? Actually, standard form is to have positive denominator, so:

(-5)^(-3) = 1 / [(-5)^3] = 1 / (-125) = -1/125

✔ Correct.

9) 12⁵ / 12¹⁰
5 - 10 = -5 → 12^(-5) = 1 / 12⁵

10) 17¹⁴ / 17¹⁴
14 - 14 = 0 → 17⁰ = 1

11) (-14)¹² / (-14)¹⁷
12 - 17 = -5 → (-14)^(-5) = 1 / (-14)⁵
Odd exponent → negative, so this equals -1 / 14⁵, but again, unless specified, we can leave as 1/(-14)⁵. However, standard simplification would be to write with positive base and sign out front.

Actually: (-14)^(-5) = 1 / [(-14)^5] = 1 / (-14⁵) = -1 / 14⁵

But since the question says “simplify”, and doesn’t specify form, either is acceptable. But best to write as -1 / 14⁵

Wait — let me double-check:
(-14)^5 = - (14^5), so 1 / (-14^5) = -1 / 14^5 → yes.

So final answer: -1 / 14⁵

12) 20³ / 20⁶
3 - 6 = -3 → 20^(-3) = 1 / 20³

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Now, let’s list all answers clearly:

1) 1/25
2) 1/3⁷
3) 1/8
4) 1/(-7)⁸ → or 1/7⁸ (since even power makes it positive) — better to write 1/7⁸
Wait — actually, (-7)^8 = 7^8, because even power removes the negative. So we can simplify to 1/7⁸

Similarly for others:

For problems with negative bases and even exponents in denominator, we can drop the negative sign in the base if exponent is even.

Let’s re-evaluate those:

4) (-7)⁴ / (-7)¹² = (-7)^(4-12) = (-7)^(-8) = 1 / (-7)^8 = 1 / 7^8 → because (-7)^8 = 7^8

6) (-3)^5 / (-3)^11 = (-3)^(-6) = 1 / (-3)^6 = 1 / 3^6 → because (-3)^6 = 3^6

8) (-5)^4 / (-5)^7 = (-5)^(-3) = 1 / (-5)^3 = 1 / (-125) = -1/125 → here exponent is odd, so stays negative.

11) (-14)^12 / (-14)^17 = (-14)^(-5) = 1 / (-14)^5 = 1 / (-14^5) = -1 / 14^5

So corrected simplified forms:

1) 1/25
2) 1/3⁷
3) 1/8
4) 1/7⁸
5) 1/4⁴
6) 1/3⁶
7) 1/6⁸
8) -1/125
9) 1/12⁵
10) 1
11) -1/14⁵
12) 1/20³

We can write them as fractions with positive exponents, which is standard.

Final verification:

All steps follow the exponent subtraction rule. Negative exponents converted to fractions. Signs handled correctly based on whether exponent is even or odd.

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Final Answer:
1) $\frac{1}{25}$
2) $\frac{1}{3^7}$
3) $\frac{1}{8}$
4) $\frac{1}{7^8}$
5) $\frac{1}{4^4}$
6) $\frac{1}{3^6}$
7) $\frac{1}{6^8}$
8) $-\frac{1}{125}$
9) $\frac{1}{12^5}$
10) $1$
11) $-\frac{1}{14^5}$
12) $\frac{1}{20^3}$