Exponent Rules worksheet with 10 problems to simplify expressions.

Let’s go through each problem one by one, using exponent rules. We’ll simplify step by step and check our work.

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Problem 1:
$\frac{7^6}{7^4}$

Rule: When dividing powers with the same base, subtract exponents: $a^m / a^n = a^{m-n}$

So: $7^{6-4} = 7^2 = 49$

✔ Check: $7^6 = 117649$, $7^4 = 2401$, $117649 ÷ 2401 = 49$ → Correct.

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Problem 2:
$\frac{4^1}{4^0}$

Any number to the power of 0 is 1 → $4^0 = 1$

So: $\frac{4}{1} = 4$

✔ Check: Direct calculation → Correct.

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Problem 3:
$2^6 \cdot 2^{-9}$

Rule: Multiply powers with same base → add exponents: $a^m \cdot a^n = a^{m+n}$

So: $2^{6 + (-9)} = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

✔ Check: $2^6 = 64$, $2^{-9} = 1/512$, $64 × (1/512) = 64/512 = 1/8$ → Correct.

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Problem 4:
$((-4)^9)^9$

Rule: Power of a power → multiply exponents: $(a^m)^n = a^{m·n}$

So: $(-4)^{9×9} = (-4)^{81}$

Since 81 is odd, negative base stays negative → answer is $-4^{81}$? Wait — actually, we write it as $(-4)^{81}$, which is a huge negative number, but we leave it in exponential form unless asked to compute.

But note: sometimes people confuse this with $-(4^{81})$. But here, since the base is -4 and raised to an odd power, it's negative. However, for simplification purposes, we just write $(-4)^{81}$.

Actually, let’s double-check: The rule applies directly: $((-4)^9)^9 = (-4)^{81}$

✔ No further simplification needed — that’s the simplified form.

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Problem 5:
$\frac{3^2}{3^3}$

Same base → subtract exponents: $3^{2-3} = 3^{-1} = \frac{1}{3}$

✔ Check: $9 / 27 = 1/3$ → Correct.

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Problem 6:
$(-4)^7 \cdot (-8)^7$

Different bases! But both have exponent 7 → use rule: $a^n \cdot b^n = (a·b)^n$

So: $[(-4) · (-8)]^7 = (32)^7$

Because negative times negative is positive.

✔ Check: Each term is negative, multiplied together → positive. And 4×8=32 → Correct.

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Problem 7:
$\frac{(-2)^{-8}}{(-2)^{-1}}$

Same base → subtract exponents: $(-2)^{-8 - (-1)} = (-2)^{-7}$

Which equals $\frac{1}{(-2)^7} = \frac{1}{-128} = -\frac{1}{128}$

Wait — let’s be careful:

Exponent subtraction: $-8 - (-1) = -8 + 1 = -7$

So: $(-2)^{-7} = \frac{1}{(-2)^7} = \frac{1}{-128} = -\frac{1}{128}$

✔ Check: Numerator: $(-2)^{-8} = 1/256$, Denominator: $(-2)^{-1} = -1/2$, so division: $(1/256) ÷ (-1/2) = (1/256) × (-2) = -2/256 = -1/128$ → Correct.

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Problem 8:
$(-8)^2 \cdot (-8)^{-5}$

Same base → add exponents: $(-8)^{2 + (-5)} = (-8)^{-3} = \frac{1}{(-8)^3} = \frac{1}{-512} = -\frac{1}{512}$

✔ Check: $(-8)^2 = 64$, $(-8)^{-5} = 1/(-32768)$, so 64 × (1/-32768) = -64/32768 = -1/512 → Correct.

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Problem 9:
$(-5)^{-4} \cdot 3^{-4}$

Different bases, same exponent → combine: $[(-5) · 3]^{-4} = (-15)^{-4}$

Now, $(-15)^{-4} = \frac{1}{(-15)^4} = \frac{1}{50625}$ because even power makes it positive.

Alternatively, you can write as $\left(\frac{1}{-15}\right)^4 = \left(\frac{1}{15}\right)^4 = \frac{1}{50625}$

✔ Check: $(-5)^{-4} = 1/625$, $3^{-4} = 1/81$, product = 1/(625×81) = 1/50625 → Correct.

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Problem 10:
$(6^{-9})^1$

Power of 1 doesn’t change anything → $6^{-9} = \frac{1}{6^9}$

We can leave it like that or compute denominator if needed, but usually left as fraction.

$6^9 = 10077696$, so $\frac{1}{10077696}$, but better to keep as $6^{-9}$ or $\frac{1}{6^9}$

The problem says “simplify”, so either is acceptable, but typically we write without negative exponents.

So final answer: $\frac{1}{6^9}$

✔ Check: Raising to power 1 does nothing → Correct.

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Final Answers Summary:

1. 49
2. 4
3. 1/8
4. $(-4)^{81}$
5. 1/3
6. $32^7$
7. -1/128
8. -1/512
9. 1/50625
10. 1/6⁹

But wait — Problem 4: Should we write it as $-4^{81}$? No! Because $(-4)^{81}$ means the entire -4 is raised to 81st power. Since 81 is odd, result is negative, but writing $-4^{81}$ would mean -(4^81), which is different. So we must keep parentheses: $(-4)^{81}$

Similarly, Problem 6: $32^7$ is correct.

Problem 9: $(-15)^{-4} = \frac{1}{(-15)^4} = \frac{1}{50625}$ — yes.

All verified.

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Final Answer:
1. 49
2. 4
3. $\frac{1}{8}$
4. $(-4)^{81}$
5. $\frac{1}{3}$
6. $32^7$
7. $-\frac{1}{128}$
8. $-\frac{1}{512}$
9. $\frac{1}{50625}$
10. $\frac{1}{6^9}$