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 = 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
