Laws of Exponents

We can write large numbers in a shorter form using exponents.

Observe 1000 = 10 × 10 × 10 = 10³

The short notation 10³ stands for the product 10×10×10. 

Here ‘10’ is called the base and ‘3’ the exponent. 

The number 10³ is read as 10 raised to the power of 3 or simply as third power of 10. 

10³ is called the exponential form of 1000.

Multiplying Powers with the Same Base:

Let us calculate 2² × 2³

2² × 2³ = (2 × 2) × (2 × 2 × 2)

             = 2 × 2 × 2 × 2 × 2

             = 2⁵ or 2²⁺³

Note:
That the Base in 2² and 2³ is same and the sum of the exponents, i.e., 2 and 3 is 5.

a² × a⁴ = (a × a) × (a × a × a × a)

         
            = a × a × a × a × a × a

            = a⁶

Note:

Base are same and the sum of the exponents is 2 + 4 = 6.

From this we can generalize that for any non-zero integer a where, m and n are whole numbers,

a^m × aⁿ =a^m+n

Simplify 3⁷ × 3⁴

Use the formula , a^m × aⁿ =a^m+n

3⁷ × 3⁴ = 3⁷⁺⁴

            =3¹¹

Dividing Powers with the Same Base:

In general, for any non-zero integer a,

a^m ÷ aⁿ =a^m-n

where m and n are whole numbers and m > n.

Simplify 3⁷ ÷ 3⁴

Use the formula , a^m ÷ aⁿ =a^m-n

3⁷ ÷ 3⁴ = 3^7-4

=3³

Power of a Power:

The generalize form for any non-zero integer ‘a’, where ‘m’ and ‘n’ are whole numbers,

 (a^m)ⁿ =a^mn 

Simplify: (5²)³

Using formula, (a^m)ⁿ =a^mn

(5²)³ = 5⁶

       =5⁶

Multiplying Powers with the Same Exponents

In general, for any non-zero integer a

a^m × b^m =(a×b)^m  (where m is any whole number)

Simplify :3² × 5²

              =(3 × 5)²

              = (15)²

Numbers with exponent 0:

Any number (except 0) raised to the power (or exponent) 0 is 1.

Simplify 3⁷ ÷ 3⁷

Use the formula , a^m ÷ aⁿ =a^m-n

3⁷ ÷ 3⁴ = 3^7-4

            =3⁰

            =1

EXAMPLE