Chapter 2. Exponents and Radicals

MODULE - 1
Algebra
Exponents and Radicals
2.4 NEGATIVE INTEGERS AS EXPONENTS
Notes
i) We know that the reciprocal of 5 is 5
1 . We write it as 5
–1
and read it as 5 raised to
power –1.
ii)
The reciprocal of (–7) is
power –1.
1
− . We write it as (–7)
7
–1 and read it as (–7) raised to the
1
iii) The reciprocal of 5 2 = 2 . We write it as 5
–2
and read it as ‘5 raised to the power (–2)’.
5
From the above all, we get
If a is any non-zero rational number and m is any positive integer, then the reciprocal of a m
⎛ 1 ⎞
⎜i. e.
m
⎟ is written as a
⎝ a
–m and is read as ‘a raised to the power (–m)’. Therefore,
⎠
1
a
m
= a
−m
Let us consider an example.
Example 2.14: Rewrite each of the following with a positive exponent:
Solution:
⎛ 3 ⎞
(i) ⎜ ⎟
⎝ 8 ⎠
−2
⎛ 4 ⎞
(ii) ⎜ − ⎟
⎝ 7 ⎠
−7
(i)
⎛ 3 ⎞
⎜ ⎟
⎝ 8 ⎠
−2
1
=
⎛ 3 ⎞
⎜ ⎟
⎝ 8 ⎠
2
=
1
2
3
2
8
8
=
3
2
2
⎛ 8 ⎞
= ⎜ ⎟
⎝ 3 ⎠
2
(ii)
⎛
⎜ −
⎝
4 ⎞
⎟
7 ⎠
−7
1
=
⎛ 4 ⎞
⎜ − ⎟
⎝ 7 ⎠
7
=
7
7
( − 4)
From the above example, we get the following result:
7
7
⎛ 7 ⎞ = ⎜ − ⎟
⎝ 4 ⎠
If q
p
is any non-zero rational number and m is any positive integer, then
⎛
⎜
⎝
p ⎞
⎟
q ⎠
−m
=
q
p
m
m
⎛
= ⎜
⎝
q
p
m
⎞
⎟ .
⎠
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Mathematics Secondary Course