Module-1 Algebra - Mathematics Secondary Course - Exponents And Radicals Worksheet Page 13
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37MODULE -
1
Exponents and Radicals
Algebra
2.5 LAWS OF EXPONENTS FOR INTEGRAL EXPONENTS
After giving a meaning to negative integers as exponents of non-zero rational numbers, we
can see that laws of exponents hold good for negative exponents also.
Notes
For example.
−
4
3
3
– 3
4
⎛
⎞
⎛
⎞
⎛
⎞
3
3
1
3
3
×
=
×
=
⎜
⎟
⎜
⎟
⎜
⎟
(i)
4
⎝
⎠
⎝
⎠
⎝
⎠
⎛
⎞
5
5
5
5
3
⎜
⎟
⎝
⎠
5
−
−
−
−
2
3
2
3
⎛
⎞
⎛
⎞
⎛
⎞
2
2
1
1
1
2
−
×
−
=
×
=
=
−
⎜
⎟
⎜
⎟
⎜
⎟
(ii)
+
2
3
2
3
⎝
⎠
⎝
⎠
⎝
⎠
⎛
⎞
⎛
⎞
⎛
⎞
3
3
3
2
2
2
−
−
−
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
3
3
3
−
−
−
3
7
7
7
3
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
3
3
1
1
1
3
3
−
÷
−
=
÷
=
×
−
=
−
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
(iii)
3
7
3
⎝
⎠
⎝
⎠
⎝
⎠
⎝
⎠
⎛
⎞
⎛
⎞
⎛
⎞
4
4
4
4
3
3
3
−
−
−
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
4
4
4
3
3
⎛
−
⎞
⎡
⎤
−
−
×
2
2
6
6
2
3
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
2
7
7
2
2
⎜
⎟
=
=
=
=
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎢
⎥
(iv)
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
⎝
⎠
⎝
⎠
7
⎢
2
⎥
2
7
7
⎝
⎠
⎣
⎦
Thus, from the above results, we find that laws 1 to 5 hold good for negative exponents
also.
∴ For any non-zero rational numbers a and b and any integers m and n,
m
n
m+n
1. a
× a
= a
÷ a
m
n
m–n
2. a
= a
if m > n
n–m
= a
if n > m
m
n
mn
3. (a
)
= a
m
m
m
4. (a × b)
= a
× b
CHECK YOUR PROGRESS 2.4
−
2
⎛ −
⎞
p
3
⎜
⎟
1. Express
as a rational number of the form
:
⎝
⎠
q
7
Mathematics Secondary Course
51
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