This worksheet helps students practice the laws of exponents through various simplification and calculation problems.

I'll solve all the problems on this worksheet step by step, working through each section carefully.

## Section A: Find the values

1) 3⁰
Any number raised to the power of 0 equals 1.
3 = 1

2) (-2)³
(-2)³ = (-2) × (-2) × (-2)
= 4 × (-2)
= -8

3) (-1)²⁰
When -1 is raised to an even power, the result is 1.
(-1)² = 1 (because 20 is even)

4) 7³
7³ = 7 × 7 × 7
= 49 × 7
= 343

5) (-5)³ + (-4)²
First: (-5)³ = (-5) × (-5) × (-5) = 25 × (-5) = -125
Second: (-4)² = (-4) × (-4) = 16
So: -125 + 16 = -109

6) (3/4)²
(3/4)² = 3²/4² = 9/16

## Section B: Simplify

1) 5² × 5⁴
When multiplying with same base, add exponents: 2 + 4 = 6
= 5⁶ ✓ (already shown)

2) 9¹¹ × 9²
Add exponents: 11 + 2 = 13
= 9¹³

3) 2⁵ × 2
Remember: 2 = 2¹
Add exponents: 5 + 1 = 6
= 2⁶

4) 8⁵ × 8⁰
Remember: 8⁰ = 1
So: 8⁵ × 1 = 8⁵
(Or using exponent rule: 5 + 0 = 5)

5) 7⁵ ÷ 7³
When dividing with same base, subtract exponents: 5 - 3 = 2
= 7²

6) 12¹⁰ ÷ 12⁶
Subtract exponents: 10 - 6 = 4
= 12⁴

7) 4²¹ ÷ 4
Remember: 4 = 4¹
Subtract exponents: 21 - 1 = 20
= 4²⁰

8) 5³ ÷ 5³
Subtract exponents: 3 - 3 = 0
= 5⁰ = 1

9) 2⁻³ × 2⁸
Add exponents: -3 + 8 = 5
= 2⁵

10) 6⁴ ÷ 6¹²
Subtract exponents: 4 - 12 = -8
= 6⁻⁸

11) 15² × 15⁻⁹
Add exponents: 2 + (-9) = -7
= 15⁻⁷

12) 7¹⁰ ÷ 7¹ × 7
First: 7¹⁰ ÷ 7¹⁴ = 7^(10-14) = 7⁻⁴
Then: 7⁴ × 7 = 7⁻ × 7¹ = 7^(-4+1) = 7⁻³
= 7⁻³

## Section C: Simplify

1) 2¹⁰ / 2⁴
Subtract exponents: 10 - 4 = 6
= 2⁶

2) 8⁻¹ / 8²
Subtract exponents: -1 - 2 = -3
= 8⁻³

3) 10⁻¹² / 10⁷
Subtract exponents: -12 - 7 = -19
= 10⁻¹⁹

4) 3⁻⁶ / 3⁻¹
Subtract exponents: -6 - (-1) = -6 + 1 = -5
= 3⁻⁵

5) (3³ × 3⁵) / 3²
Numerator first: 3³ × 3⁵ = 3
Then: 3⁸ / 3² = 3^(8-2) = 3⁶
= 3⁶

6) (6⁻² × 6⁴) / 6⁻⁸
Numerator first: 6⁻² × 6⁴ = 6²
Then: 6² / 6⁻⁸ = 6^(2-(-8)) = 6^(2+8) = 6¹⁰
= 6¹⁰

7) (7⁻⁸ × 7⁻⁷) / 7⁻⁴
Numerator first: 7⁻⁸ × 7⁻⁷ = 7⁻¹⁵
Then: 7⁻¹⁵ / 7⁴ = 7^(-15-(-4)) = 7^(-15+4) = 7⁻¹¹
= 7⁻¹¹

8) 14⁻³ / (14⁻¹¹ × 14⁻¹)
Denominator first: 14⁻¹¹ × 14⁻¹⁹ = 14⁻³⁰
Then: 14⁻³ / 14⁻³⁰ = 14^(-3-(-30)) = 14^(-3+30) = 14²⁷
= 14²⁷

9) (4⁸ × 4⁵) / (4² × 4⁴)
Numerator: 4⁸ × 4⁵ = 4¹³
Denominator: 4² × 4⁴ = 4⁶
Then: 4¹³ / 4⁶ = 4^(13-6) = 4⁷
= 4⁷

10) (6⁻⁷ × 6³) / (6⁻¹ × 6)
Numerator: 6⁻⁷ × 6³ = 6⁻⁴
Denominator: 6⁻¹ × 6¹ = 6⁰ = 1
Then: 6⁴ / 1 = 6⁻
= 6⁻⁴

11) (20⁻¹⁷ × 20⁹) / (20 × 20⁻¹¹)
Numerator: 20⁻¹⁷ × 20⁹ = 20⁻⁸
Denominator: 20 × 20⁻¹¹ = 20⁻⁷
Then: 20⁸ / 20⁻⁷ = 20^(-8-(-7)) = 20^(-8+7) = 20⁻¹
= 20⁻¹

12) (5e⁸ × 4e⁵) / (2e² × 10e⁴)
Numbers: (5 × 4) / (2 × 10) = 20 / 20 = 1
Exponents: e⁸ × e = e¹³ in numerator
e² × e⁴ = e in denominator
e¹³ / e = e⁷
= e⁷ or 1e⁷

## Extension Section

Investigate the following:

(2¹)² = 2¹ × 2¹ = 2² ✓ (shown)

(2²)² = 2² × 2² = 2⁴ = 16

(2³)² = 2³ × 2³ = 2⁶ = 64

(2⁴)² = 2⁴ × 2 = 2⁸ = 256

(2⁵)² = 2⁵ × 2⁵ = 2¹⁰ = 1024

(5¹)³ = 5¹ × 5¹ × 5¹ = 5³ ✓ (shown)

(5²)³ = 5² × 5² × 5² = 5⁶ = 15,625

(5³)³ = 5³ × 5³ × 5³ = 5⁹ = 1,953,125

(5⁴)³ = 5 × 5⁴ × 5 = 5¹² = 244,140,625

(5⁵)³ = 5⁵ × 5⁵ × 5⁵ = 5¹⁵ = 30,517,578,125

Use your results to simplify:

Pattern observed: When you have (a)ⁿ, it equals a^(m×n)

(4²)⁴ = 4^(2×4) = 4 = 65,536

(6³)⁸ = 6^(3×8) = 6²⁴ = 473,838,133,832,161,689,6

(9⁻⁵)⁶ = 9^(-5×6) = 9⁻³⁰

(11⁻²)⁻⁷ = 11^(-2×-7) = 11¹ = 379,749,833,583,241

──────────────────────────────────────

Final Answer:

Section A:
1) 1
2) -8
3) 1
4) 343
5) -109
6) 9/16

Section B:
1) 5⁶
2) 9¹³
3) 2⁶
4) 8⁵
5) 7²
6) 12⁴
7) 4²⁰
8) 1
9) 2⁵
10) 6⁻⁸
11) 15⁻⁷
12) 7⁻³

Section C:
1) 2⁶
2) 8⁻³
3) 10⁻¹⁹
4) 3⁻⁵
5) 3⁶
6) 6¹⁰
7) 7⁻¹¹
8) 14²⁷
9) 4
10) 6⁻⁴
11) 20⁻¹
12) e⁷

Extension - Investigate:
(2²)² = 2⁴ = 16
(2³)² = 2⁶ = 64
(2⁴)² = 2⁸ = 256
(2⁵)² = 2¹⁰ = 1024
(5²)³ = 5⁶ = 15,625
(5³)³ = 5⁹ = 1,953,125
(5⁴)³ = 5¹² = 244,140,625
(5⁵)³ = 5¹⁵ = 30,517,578,125

Extension - Use results to simplify:
(4²) = 4⁸ = 65,536
(6³)⁸ = 6²⁴
(9⁻)⁶ = 9⁻³
(11⁻²)⁻⁷ = 11¹⁴