Indices

Rules for same base

a^m x aⁿ = a^m+n

5³ x 5² = 5³⁺² = 5⁵

a^m ÷ aⁿ = a^m-n

4⁵ ÷ 4³ = 4^5-3 = 4²

Rules for same index

a^m x b^m = (ab)^m

5³ x 2³ = (5×2)³ = 10³

a^m ÷ b^m = (a÷b)^m

4³ ÷ 2³ = (4÷2)³ = 2³

(a^m)ⁿ = a^mn

(2³)² = 2^3×2 = 2⁶

Rules for negative index

a^-m = 1/ a^m

2^-5 = 1/2⁵

Rules for fractional index

√a = a^½

√9 = 9^½ = (3²)^½ = 3

Examples

Solve 2^3x = 8

2^3x = 2³, since bases are equal, powers are equal

3x = 3

x = 1

Solve 4^x = 64^1/2

4^x = (4³)^1/2

4^x = 4^3/2, since are equal, powers are equal

x = 3/2 = 1.5

Solve (2x)⁵ = 243

(2x)⁵ = 3⁵

2x = 3, since powers are equal, bases are equal

x = 1.5

Solve Z⁵ = 27 x Z²

Z⁵ ÷ Z² = 27

Z³ = 3³, since powers are equal, bases are equal

z = 3

Simplify : (a²)⁵

= a^2x5

= a¹⁰

Simplify : (2a²)⁵ x (a³)

= 2⁵ x a¹⁰ x a³

= 32 a^{10 + 3}

= 32 a¹³

Solve 2^2x = 16

2^2x = 2⁴

2x = 4 bases are equal, so powers are equal

x = 2

Solve 9^x = 27

(3²)^x = 3³

3^2x = 3³

2x = 3

x = 1.5

By making a suitable substitution, solve 5^2x – 2 = 6(5^x) – 7

5^2x – 2 = 6(5^x) – 7

(5^x)² – 2 = 6(5^x) – 7

Let y = 5^x

y² – 2 = 6y – 7

y² – 2 – 6y + 7 = 0

y² – 6y + 5 = 0

Factorising,

(y – 5) (y – 1) =0

y = 5 or y = 1

5^x = 5 or 5^x = 1

5^x = 5¹ or 5^x = 5⁰

x = 1 or x = 0

Questions

1. Find the value of 4^-3

2. Find the value of 16^1/2

3. Find the value of 4² x 4³

4. Find the value of 9⁵ / 9²

5. Find the value of 4² x 3²

6. Find the value of 9³ / 3³

7. Find the value of 4² x 2⁵

8. Find the value of 9³ / 3⁴

9. Find the value of 6² + 6¹ + 6⁰ + 6^-1

10. Solve : a² = 25

11. Solve : b³ = – 27

12. Solve : 5^2x = 125

13. Solve : 3^y = 243

14. Simplify : (2x⁴)³