## Multiplying numbers with same base

a m × a n = a m + n

Let’s take some examples & check

2 5 × 2 3   = (2 × 2 × 2 × 2 × 2) × (2 × 2 × 2)

= 2 8

∴ 2 5 × 2 3 = 2 5 + 3 = 2 8

Similarly,

(−3) 4 × (−3) 2

= (−3 × −3 × −3 × −3) × (−3 × −3)

= −3 × −3 × −3 × −3 × −3 × −3

= (−3) 6

∴ (−3) 4 (−3) 2 = (−3) 4 + 2 = (−3) 6

And for,

5 6 × 5 3 = (5 × 5 × 5 × 5 × 5 × 5) × (5 × 5 × 5)

= 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5

= 5 9

∴ 5 6 × 5 3 = 5 6 + 3 = 5 9

Dividing Numbers with same base

a m /a n  = a m - n

2 5 /2 3  = (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2)

= 2 × 2

= 2 2

∴ 2 5 /2 3 = 2 5 - 3 = 2 2

Similarly,

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

= -3 × -3

= (-3) 2

∴ (-3) 4 /(-3) 2  = (-3) 4 - 2 = (-3) 2

And for,

5 6 /5 3  = (5 × 5 × 5 × 5 × 5 × 5)/(5 × 5 × 5)

= 5 × 5 × 5

= 5 3

∴ 5 6 /5 3  = 5 6 − 3   = 5 3

### Power of a Power

Thus we can write

(a m ) n = a m × n

Suppose we have

(2 3 )  = 2 3 × 2       (As a m × a n = a m + n )

= 2 6

So, (2 3 ) 2   = 2 3 × 2 = 2 6

Similarly,

(3 2 ) 4   = 3 2 × 4

= 3 8

(7 4 ) 5   = 7 4 × 5

= 7 20

(8 9 ) 3   = 8 9 × 3

= 8 27

### Multiplying number with same power

We note that

a m × b m = (a × b) m

Suppose we have

2 2 × 3 2   = (2 × 2) × (3 × 3)

= (2 × 3) × (2 × 3)

= 6 × 6

= 6 2

Thus,

2 2 × 3 2 = (2 × 3) 2

= 6 2

Similarly,

5 3 × 7 3

= (5 × 5 × 5) × (7 × 7 × 7)

= (5 × 7) × (5 × 7) × (5 × 7)

= 35 × 35 × 35

= 35 3

∴ 5 3 × 7 3   = (5 × 7) 3

= 35 3

### Dividing number with same power

We note that

a m /b m = (a/b) m

Suppose we have,

2 2 /3 2  = (2 × 2)/(3 × 3)

= 2/3 × 2/3

= (2/3) 2

∴ 2 2 /3 2  = (2/3) 2

Similarly,

5 3 /7 3   =  (5 × 5 × 5)/(7 × 7 × 7)

= (5/7)×(5/7)×(5/7)

= (5/7) 3

∴ 5 3 /7 3  = (5/7) 3

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1. Chapter 13 Class 7 Exponents and Powers
2. Concept wise
3. Law of exponents

Law of exponents 