Indices

Index laws (exponent rules) let you simplify expressions with powers. When multiplying powers with the same base, add the indices: aᵐ × aⁿ = aᵐ⁺ⁿ. When dividing, subtract: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. A power raised to another power multiplies the indices: (aᵐ)ⁿ = aᵐⁿ. Any number to the power of 0 equals 1, and a negative index means "one over" — a⁻ⁿ = 1/aⁿ.

Key terms

Base:

The number being multiplied by itself in a power, e.g. the 2 in 2³.

Index / exponent:

The small number showing how many times to multiply the base, e.g. the 3 in 2³.

Questions

1. 1. What does 2³ mean?

   • 2 × 2 × 2
   • 2 × 3
   • 2 + 2 + 2
   • 3 × 3

2. 2. What is 2³?

   • 6
   • 8
   • 9
   • 12

3. 3. What is 5²?

   • 10
   • 25
   • 15
   • 7

4. 4. In 3⁴, what is the base?

   • 3
   • 4
   • 7
   • 12

5. 5. In 3⁴, what is the index?

   • 3
   • 4
   • 7
   • 12

6. 6. Any number to the power of 0 equals:

   • 0
   • 1
   • The number itself
   • Infinity

7. 7. What is 10¹?

   • 1
   • 10
   • 100
   • 0

8. 8. Simplify: 2³ × 2⁴

   • 2⁷
   • 2¹²
   • 4⁷
   • 2¹

9. 9. Simplify: 5⁶ ÷ 5²

   • 5⁴
   • 5³
   • 5⁸
   • 5¹²

10. 10. Simplify: (2²)³

    • 2⁵
    • 2⁶
    • 2⁹
    • 4⁶

11. 11. Simplify: 3⁵ × 3²

    • 3⁷
    • 3¹⁰
    • 9⁷
    • 3³

12. 12. Simplify: 7⁸ ÷ 7⁵

    • 7³
    • 7¹³
    • 7⁴⁰
    • 7¹

13. 13. What is 4⁻¹?

    • 1/4
    • −4
    • −1/4
    • 4

14. 14. Simplify: (x³)²

    • x⁵
    • x⁶
    • x⁹
    • x²

15. 15. Simplify: 2⁵ × 2⁻²

    • 2³
    • 2⁷
    • 2⁻¹⁰
    • 2¹⁰

16. 16. Simplify: (3x²)³

    • 9x⁶
    • 27x⁶
    • 27x⁵
    • 3x⁶

17. 17. Simplify: (2⁴ × 2³) ÷ 2⁵

    • 2²
    • 2¹²
    • 2⁷
    • 2¹

18. 18. What is 2⁻³ as a fraction?

    • 1/8
    • -8
    • 1/6
    • 8

19. 19. Simplify: x⁵ ÷ x⁻²

    • x⁷
    • x³
    • x⁻⁷
    • x¹⁰

20. 20. Simplify: (2³)⁰

    • 1
    • 0
    • 8
    • 2³

21. 21. Simplify: (a²b³)²

    • a⁴b⁶
    • a²b⁶
    • a⁴b⁵
    • a²b³