Laws of Exponents Worksheets - Free Printable

Let’s solve each problem step by step. We’ll use the rules of exponents and order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

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1. 3³ + 5²

- 3³ = 3 × 3 × 3 = 27
- 5² = 5 × 5 = 25
- Add them: 27 + 25 = 52

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2. 4⁴ + 6³

- 4⁴ = 4 × 4 × 4 × 4 = 256
- 6³ = 6 × 6 × 6 = 216
- Add them: 256 + 216 = 472

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3. 7³ × 5⁰

- Any number to the power of 0 is 1 → 5⁰ = 1
- 7³ = 7 × 7 × 7 = 343
- Multiply: 343 × 1 = 343

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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 = 3 × 3 × 3 × 3 = 81
- Multiply: -64 × 81

Let’s compute that:

64 × 80 = 5120
64 × 1 = 64
So 64 × 81 = 5120 + 64 = 5184
But since it’s -64 × 81 → -5184

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5. 8⁶ × 3⁻²

Negative exponent means reciprocal: 3⁻² = 1 / 3² = 1/9

8⁶ is big — let’s calculate:

8² = 64
8⁴ = 64 × 64 = 4096
8⁶ = 8⁴ × 8² = 4096 × 64

Compute 4096 × 64:

Break it down:
4096 × 60 = 245,760
4096 × 4 = 16,384
Add: 245,760 + 16,384 = 262,144

Now multiply by 1/9:
262,144 ÷ 9

Do division:

9 × 29,127 = 262,143 → remainder 1

So 262,144 / 9 = 29,127 and 1/9, or as a fraction: 262144/9

We can leave it as a fraction unless told otherwise.

→ 262144/9

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6. 6³ × 4⁻²

4⁻² = 1 / 4² = 1/16
6³ = 6 × 6 × 6 = 216

Multiply: 216 × (1/16) = 216 / 16

Simplify: divide numerator and denominator by 8 → 27 / 2 = 13.5 or 27/2

Either is fine, but since others are integers or fractions, we’ll write as 27/2

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7. (8x⁹)(12x³)

Multiply coefficients: 8 × 12 = 96
Multiply variables: x⁹ × x³ = x^(9+3) = x¹²

→ 96x¹²

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8. (12x⁴)(13x²)

Coefficients: 12 × 13 = 156
Variables: x⁴ × x² = x⁶

→ 156x⁶

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9. (21x²)(14x²)

Coefficients: 21 × 14 = 294
Variables: x² × x² = x⁴

→ 294x⁴

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10. (34x⁵)(19x³)

Coefficients: 34 × 19

Calculate:
34 × 20 = 680 → minus 34 = 646
Variables: x⁵ × x³ = x⁸

→ 646x⁸

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11. f(x) = (-2x)² · (4x)

First, (-2x)² = (-2)² × x² = 4x²
Then multiply by 4x: 4x² × 4x = 16x³

→ f(x) = 16x³

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12. f(x) = (22x⁵)² · (11x)²

First, square each part:

(22x⁵)² = 22² × (x⁵)² = 484 × x¹⁰
(11x)² = 121 × x²

Now multiply: 484x¹⁰ × 121x²

Coefficients: 484 × 121

Let’s compute:

Note: 484 = 22², 121 = 11² → so 484 × 121 = (22×11)² = 242²? Wait, no — actually:

Better to compute directly:

484 × 100 = 48,400
484 × 20 = 9,680
484 × 1 = 484
Add: 48,400 + 9,680 = 58,080 + 484 = 58,564

Variables: x¹⁰ × x² = x¹²

→ f(x) = 58564x¹²

Wait — double-check 484 × 121:

Alternative way:
484 × 121 = 484 × (100 + 20 + 1) = same as above → 58,564 ✔️

Or note: 22² × 11² = (22×11)² = 242²

242² = ?

240² = 57,600
2×240×2 = 960
2² = 4
Total: 57,600 + 960 + 4 = 58,564 ✔️

Yes.

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Final Answers:

1. 52
2. 472
3. 343
4. -5184
5. 262144/9
6. 27/2
7. 96x¹²
8. 156x⁶
9. 294x⁴
10. 646x⁸
11. 16x³
12. 58564x¹²

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Final Answer:
1. 52
2. 472
3. 343
4. -5184
5. \frac{262144}{9}
6. \frac{27}{2}
7. 96x^{12}
8. 156x^6
9. 294x^4
10. 646x^8
11. 16x^3
12. 58564x^{12}