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2
4
6
6
2 2
2
2 × ( 3 ) × 3 × 2 8 2 × 3 × 3 × 2 8
m
m
m
m n
mn
= = [Q(a ) = a , (ab) = a × b ]
3
6
4
3
2 ( 23 × 3 × 2 × 3 4 2 × 3 × 2 × 3 4
4
)
2 28 × 3 46 2 × 3 10
+
10
+
m
n
= = [Q a × a = a m+n ]
2 64 × 3 34 2 × 3 7
10
+
+
3
m
n
0
= 2 10 – 10 × 3 10 – 7 = 2 × 3 = 1 × 27 = 27 [Q a ÷ a = a m–n ]
(
4
2 × a 3 × 5a 4 22×× 2 × 5 × a 3 × a ) 22×× a 3+4
3
n
m
(b) = = [Q a × a = a m + n ]
10 a 2 2 × 5 × a 2 a 2
22×× a 7
n
−
4
m
= =× a 72 = a 4 5 [Q a ÷ a = a m – n ]
a 2
4
6
Example 18: By what number should (–2) be multiplied so that the product is equal to (4) ?
6
Solution: To get the required number, divide the (4) by (–2) . Therefore,
4
4
The required number = (4) ÷ (–2) 6
2 4
= (2 ) ÷ (–2) [Q (a ) = a ]
m n
6
mn
8
6
= (2) ÷ (2) 6 [Q (–1) = 1, as 6 is even number]
= 2 8 – 6
= 2 2
Thus, the required number = 4
Example 19: By what number should 5 be divided to obtain 625?
25
Solution: To get the required number divide 5 by 625. Therefore,
25
25
The required number = 5 ÷ 625
= 5 ÷ 5 4
25
= 5 25 – 4 = 5 21
Thus, the required number is 5 .
21
Enrichment
If on both the sides, powers have the same base. So, their exponents must be equal.
Let us take an example based on the above statement.
5

5
5
5
Find x so that    ×    11 =   8x
 
 
 
3
3
3
3

5
5
5
Given,    ×    11 =   8x
 
 
 
3
3
3
5

m
a

So, 5 5 × 5 11 =   8x  Using   m = a 


3
3 5 3 11      b 
b
m
×
5
55 11  5 8x (5) 16  5 8x
m
n
or =   or =   [Using a × a = (a) m + n ]
×
5
3
33 11   (3) 16  
3
5
5

or    16 =   8x or 16 = 8x
 
 
Thus, 8x = 16 3 3
Therefore, x = 2
3

2
2
2
Find m so that    ×    6 =   2m-1
 
 
 
9
9
9
89 Exponents and Powers