Module-1 Algebra - Mathematics Secondary Course - Exponents And Radicals Worksheet Page 15
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37MODULE -
1
Exponents and Radicals
Algebra
1
In other words, the qth power of
or
=
q
a
a
1
q
in other words
is the qth root of a and is written as
a .
Notes
q
a
For example,
1
1
1
1
1
1
1
1
4
+
+
+
×
×
×
=
=
=
=
1
4
4
4
4
4
4
4
4
4
7
7
7
7
7
7
7
7
1
or
7 is the fourth root of 7 and is written as
7 ,
4
4
Let us now define rational powers of a
If a is a positive real number, p is an integer and q is a natural number, then
p
a =
q
q
p
a
We can see that
p
p
p
p
p
p
p
p
+
+
+
.....q tim
es
.
q
×
×
×
=
=
=
q
q
q
q
q
q
q
q
p
a
a
a
........
a
a
a
a
q times
p
∴
a =
q
p
q
a
∴ a
p/q
p
is the qth root of a
2
2
Consequently,
7 is the cube root of 7
.
3
Let us now write the laws of exponents for rational exponents:
m
n
m+n
(i) a
× a
= a
÷ a
m
n
m–n
(ii) a
= a
m
n
mn
(iii) (a
)
= a
m
m
m
(iv) (ab)
= a
b
m
⎛
⎞
m
a
a
=
⎜
⎟
(v)
⎝
⎠
m
b
b
Let us consider some examples to verify the above laws:
Example 2.15:
Find the value of
−
3
4 /
⎛
⎞
16
1
2
⎜
⎟
(
)
(
)
(i)
(ii)
(iii)
4
5
625
243
⎝
⎠
81
Mathematics Secondary Course
53
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