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

  1. Chapter 13 Class 7 Exponents and Powers
  2. Concept wise

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.