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Get it right!

5
6
5
51
3
49
(–1) × (–1) = (–1) 51 × 49 3 × 5 = 3 5 + 3 3 × 3 = 3 5 + 6
51
5
(–1) × (–1) = (–1) 51 + 49 = (–1) 100 3 × 2 = 2 5 + 5
49
5
Law II: Dividing Powers with the Same Base
Let us take some examples,
×××
(a) 2 5 = 222 2 × 2 =× ×= 2 = 2 5 2
3
−
222
2 2 2 × 2
3
) (−×
(b) (−3 ) 5 = (−× ) 3 (−3 ) × (−3 ) × (−3 ) =− ×−3))(=−3 ) =−3 ) 53
2
(
−
( 3
) (
(−3 ) 3 (−3 ) × (−3 ) × (−3 )
a × a × a × a

−
2
2
4
Let a be a non-zero integer, then a ÷ a = = a × a = a = a 4 2

a × a
Therefore, a ÷ a = a 4 – 2 = a 2
4
2

From the above examples, it is clear that if we divide a quantity by another quantity having the
same base, then the quotient thus obtained by subtracting the exponent of the denominator from
the exponent of the numerator, whereas the base remains the same.
In general way, if a be a non-zero integer and m, n are two whole numbers, then a ÷ a =
n
m
a m = a m – n .
a n
It is called II law of exponents.
Example 10: Simplify the following in exponential form:
4
7
8
14
15
13
(a) 3 ÷ 3 (b) 8 ÷ 8 4 (c) (–10) ÷ (–10) (d) (–x) ÷ (–x) 10
m
7
4
n
Solution: (a) 3 ÷ 3 = 3 7 – 4 = 3 3 (Q a ÷ a = a m – n )
8
n
m
4
(b) 8 ÷ 8 = 8 8 – 4 = 8 4 (Q a ÷ a = a m – n )
13
m
n
15
(c) (–10) ÷ (–10) = (–10) 15 – 13 = (–10) 2 (Q a ÷ a = a m – n )
m
n
14
10
(d) (–x) ÷ (–x) = (–x) 14 – 10 = (–x) 4 (Q a ÷ a = a m – n )
Law III: Taking Power of a Power
Let us try to simplify (2 ) and (3 ) .
3 2
2 4
3 2
3
Now, (2 ) means 2 is multiplied two times with itself.
m
n
3 2
3
3
(2 ) = 2 × 2 = 2 3 + 3 (Q a × a = a m + n )
= 2 = 2 3 × 2 (6 is the product of 3 and 2).
6
Thus, (2 ) = 2 3 × 2
3 2
2 4
2
Similarly, (3 ) = 3 × 3 × 3 × 3 2
2
2
n
= 3 2 + 2 + 2 + 2 (Q a × a = a m + n )
m
8
= 3 = 3 2 × 4 (8 is the product of 2 and 4).
Mathematics-7 84