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It is clear from the above examples that if we multiply two quantities having different bases but the
same exponent, then the product is obtained by multiplying their bases, keeping the exponent same.

m
m
m
In general, for two non-zero integers a and b : a × b = (ab) , where m is any whole number.
It is called the IV law of exponents.

Maths Talk

For any two numbers, which usually gives the greater number, either using the greater number as the base or
the exponent? Why? Discuss with your classmate. Are there any exception cases? If yes, then give at least one
exception.

Example 12: Simplify the following in exponential form:

4
(a) 2 × 3 (b) (–3) × (–2) 2
4
2
3
6
6
(c) (–2) × 5 (d) (–x) × (–y) 3
m
m
4
4
4
m
Solution: (a) 2 × 3 = (2 × 3) = 6 4 [Q a × b = (ab) ]
m
2
m
2
2
m
(b) (–3) × (–2) = {(–3) × (–2)} = 6 2 [Q a × b = (ab) ]
m
6
6
6
m
m
(c) (–2) × 5 = {(–2) × 5} = (–10) 6 [Q a × b = (ab) ]
m
m
m
3
3
3
(d) (–x) × (–y) = [(–x) × (–y)] = (xy) 3 [Q a × b = (ab) ]
Example 13: Express the following terms in the exponential form:
(a) (4 × 9) 3 (b) (2n) 4 (c) (–4a) 3
3
Solution: (a) (4 × 9) = (4 × 9) × (4 × 9) × (4 × 9)
= (4 × 4 × 4) × (9 × 9 × 9)
= 4 × 9 3
3
4
4
(b) (2n) = (2 × n)
= (2 × n) × (2 × n) × (2 × n) × (2 × n)
= (2 × 2 × 2 × 2) × (n × n × n × n)
4
= 2 × n 4
3
(c) (– 4a) = (– 4 × a) 3
= (– 4 × a) × (– 4 × a) × (– 4 × a)
= (– 4) × (– 4) × (– 4) × (a × a × a)
= (– 4) × (a) 3
3
Law V: Dividing Powers with the Same Exponents
Observe the following examples:
×××
2 4 2222 2 2 2 2  2 4
(a) = = × × × =  
3
3 4 3333 3 3 3 3  
×××
××
a 3 aaa a a a  a 3
(b) = = × × =  
b
××
b 3 bbb b b b  
Mathematics-7 86