Laws of Indices

 

The following laws of indices are true for all values of a, b and x not equal to 0 (zero)

·       X^a x X^b = X^a+b

·       X^a x X^b = X^a-b

·       X⁰ = 1, where X is not equal 0 ( X ≠ 0)

·       X^-a = 1/X^-a

Example

Simplify (a) 3⁵ X 3⁴    (b) 12X⁴ ÷ 3X² (c) 4⁰ (d) 2^-4

Solutions

a) 3⁵ x 3⁴ = 3⁵⁺⁴ = 3⁹

We know that (3x3x3x3x3) = 3⁵ and (3x3x3x3) = 3⁴ so multiplying both

(3x3x3x3x3) x (3x3x3x3) so if you remove the brackets, you will have

3x3x3x3x3x3x3x3x3 = 3⁹

 

b) 12x⁴ ÷ 3x²

  =12 /3 x X^4-2

   = 4 x X^4-2

= 4x²

 

c) 4⁰  = 1

Since x-x = 0

Then 4⁰ = 4^x ÷ 4^x = 1

 

d) 2^-4  = ½⁴  = 1/16

 

Product of indices (x^a)^b

Generally,

(X^a)^b = X^axb = X^ab

 

Simplify the following indices

a) (X³)⁴ 

b) (a²)⁴

c) (Q²)⁵

 

Solutions

a) (X³)⁴   = X³ x X³ x X³ x X³ =  X¹²

 Or   (X³)⁴ = X^3x4 = X¹²

b) (a²)⁴ = a² x a² x a² x a²  = a⁸

 

Or

(a²)⁴ = a^2x4 = a⁸

 

 

c) (Q²)⁵ = Q² x Q² x Q² x Q² x Q²  = Q¹⁰

 

Or

 

 

(Q²)⁵ = Q^2x5 = Q¹⁰

 

Fractional Indices

 

√a is a short form of square root of a. so,

√ a x √ a = √ a x a = a.

So, let

√a = a^x

Then a^x x a^x = √a x √a = a¹

 

a^x x a^x  = a¹

a^x+x  = a¹

a^2x = a¹  

if you cancel a. from both sides of the equation

2x = 1

X = ½

 

Note

√a = a^x

 

Since x = ½ then

 √a =  X^1/2 

Similarly,

3√a is a short forum of cube root of a.

 

3√a x 3√a x 3√a = a

 Let 3√a = a^p

 

Then a^p x a^p x a^p = a¹

 

a.^3p = a¹

Cancel a from both sizes of the equation

3p = 1

P = 1/3

3√a = a^1/3