Negative Exponents Worksheets with Answer Key - Free Printable

Let’s solve each problem step by step. We’ll go one at a time, carefully.

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1. 3³ + 5²

3³ = 3 × 3 × 3 = 27
5² = 5 × 5 = 25
27 + 25 = 52

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2. 4⁴ + 6³

4⁴ = 4 × 4 × 4 × 4 = 256
6³ = 6 × 6 × 6 = 216
256 + 216 = 472

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3. 7³ + 5⁴

7³ = 7 × 7 × 7 = 343
5⁴ = 5 × 5 × 5 × 5 = 625
343 + 625 = 968

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4. -4³ × 3⁴

Important: The negative sign is NOT part of the base unless in parentheses. So -4³ means -(4³), not (-4)³.

4³ = 64 → so -4³ = -64
3⁴ = 81
-64 × 81 = let’s compute:

64 × 80 = 5120
64 × 1 = 64
So 64 × 81 = 5184 → then make it negative: -5184

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5. 8⁴ × 3⁻²

First, 8⁴ = (2³)⁴ = 2¹² = 4096
But maybe easier to compute directly:
8² = 64 → 8⁴ = 64 × 64 = 4096

3⁻² = 1 / 3² = 1/9

So 4096 × (1/9) = 4096 ÷ 9

Let’s divide:
9 × 455 = 4095 → remainder 1 → so 455 and 1/9 → as improper fraction: 4096/9

We can leave it as a fraction since it doesn’t simplify.

Answer: 4096/9

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6. 6⁻³ × 4⁻³

Negative exponents mean reciprocal.

6⁻³ = 1/(6³) = 1/216
4⁻³ = 1/(4³) = 1/64

Multiply: (1/216) × (1/64) = 1/(216×64)

Compute 216 × 64:

200 × 64 = 12,800
16 × 64 = 1,024
Total = 12,800 + 1,024 = 13,824

So answer: 1/13824

Alternatively, notice both have exponent -3:

(6 × 4)⁻³ = 24⁻³ = 1/(24³)

24² = 576 → 24³ = 576 × 24

576 × 20 = 11,520
576 × 4 = 2,304
Total = 13,824 → same result.

Answer: 1/13824

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7. (8x)⁰(12x)³

Anything to power 0 is 1 → (8x)⁰ = 1

(12x)³ = 12³ × x³ = 1728x³

So 1 × 1728x³ = 1728x³

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8. (12x)²(13x)²

Use rule: (ab)ⁿ = abⁿ

= (12²)(x²)(13²)(x²) = (144)(169)(x⁴)

Now multiply 144 × 169

Break it down:

144 × 170 = 144 × 100 + 144 × 70 = 14,400 + 10,080 = 24,480
Then subtract 144 → 24,480 - 144 = 24,336

So answer: 24336x⁴

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9. (21x)²(14x)³

First, expand:

(21x)² = 21² × x² = 441x²
(14x)³ = 14³ × x³ = 2744x³

Now multiply: 441 × 2744 × x^(2+3) = 441 × 2744 × x⁵

This is big — let’s compute 441 × 2744.

Note: 441 = 21², 2744 = 14³ = (2×7)³ = 8×343 = but maybe better to factor.

Actually, let’s write everything in prime factors to simplify before multiplying.

21 = 3×7 → 21² = 3²×7²
14 = 2×7 → 14³ = 2³×7³

So total coefficient: 3² × 7² × 2³ × 7³ = 2³ × 3² × 7⁵

Compute:

2³ = 8
3² = 9
7⁵ = 7×7=49; 49×7=343; 343×7=2401; 2401×7=16807

Wait — that’s too big? Let me check:

Actually, we can compute numerically:

441 × 2744

Let me do 400 × 2744 = 1,097,600
40 × 2744 = 109,760
1 × 2744 = 2,744
Add them: 1,097,600 + 109,760 = 1,207,360 + 2,744 = 1,210,104

And x⁵

So answer: 1210104x⁵

Wait — let me verify with another method.

21² = 441
14³ = 14×14=196; 196×14=2744 ✔️

Now 441 × 2744:

Do 441 × 2700 = 441 × 27 × 100

441 × 20 = 8820
441 × 7 = 3087
Sum: 8820 + 3087 = 11907 → times 100 = 1,190,700

Then 441 × 44 = ?

441 × 40 = 17,640
441 × 4 = 1,764
Sum: 19,404

Total: 1,190,700 + 19,404 = 1,210,104 ✔️

Yes.

Answer: 1210104x⁵

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10. (34x)²(19x)³

Expand:

(34x)² = 34² × x² = 1156x²
(19x)³ = 19³ × x³ = 6859x³

Multiply coefficients: 1156 × 6859 × x⁵

This is very large — perhaps leave as is? But worksheet says “simplify” — probably expects numerical coefficient.

Let’s compute 1156 × 6859.

Too big for mental math easily — maybe use calculator-style steps.

But since this is a student worksheet, perhaps they expect us to leave it factored? No — instruction says “simplify”, and others are computed.

Wait — maybe I made a mistake? Let me see if there's a pattern or if numbers are chosen to be nice.

34 and 19 — no common factors.

Perhaps compute step by step.

Alternative: Use prime factorization? Not helpful here.

Maybe accept that it’s big.

But let’s try:

First, 1156 × 6000 = 6,936,000
1156 × 800 = 924,800
1156 × 50 = 57,800
1156 × 9 = 10,404

Now add:

Start with 6,936,000
+924,800 = 7,860,800
+57,800 = 7,918,600
+10,404 = 7,929,004

Is that right?

Check: 6859 = 6000 + 800 + 50 + 9? 6000+800=6800; +50=6850; +9=6859 ✔️

Now:

1156 × 6000 = 1156 × 6 × 1000 = 6936 × 1000 = 6,936,000 ✔️
1156 × 800 = 1156 × 8 × 100 = 9248 × 100 = 924,800 ✔️
1156 × 50 = 1156 × 5 × 10 = 5780 × 10 = 57,800 ✔️
1156 × 9 = let's compute: 1000×9=9000; 156×9=1404; total 10,404 ✔️

Sum:
6,936,000
+ 924,800 = 7,860,800
+ 57,800 = 7,918,600
+ 10,404 = 7,929,004

Yes.

So answer: 7929004x⁵

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11. f(x) = (-2x)² ÷ (4x)

First, (-2x)² = (-2)² × x² = 4x²

Divide by (4x): 4x² ÷ 4x = (4÷4) × (x²÷x) = 1 × x = x

So f(x) = x

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12. f(x) = (22x)⁰ ÷ (11x)²

(22x)⁰ = 1 (anything to 0 is 1, as long as not zero — assuming x ≠ 0)

(11x)² = 121x²

So 1 ÷ 121x² = 1/(121x²)

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Now, compiling all answers:

Final Answer:
1. 52
2. 472
3. 968
4. -5184
5. 4096/9
6. 1/13824
7. 1728x³
8. 24336x⁴
9. 1210104x⁵
10. 7929004x⁵
11. x
12. 1/(121x²)