All of you might be aware of Integer Exponents . Let’s get into a little tougher concept i.e. Rational Exponents. Usually, Rational Exponent can be expressed in the form of (b) m/n where m, n are integers. In Rational Exponents, there are two types namely Positive Rational Exponent and Negative Rational Exponent. Have a glance at the solved examples explaining the concept and get a grip on it and learn how to solve the related problems.
Positive Rational Exponent
Let us consider x and y to be non zero rational numbers and m is a positive integer such that x m = y then we can express it in the form of x= (y) 1/m . However, we can write y 1/m = m√y and is referred to as the mth root of y.
y
1/3
= 3√y, y
1/5
= 5√y, etc. Consider a positive rational number x having the rational exponent p/q then x can be represented in the following fashion.
X
(p/q)
= (x
p
)
1/q
= q√x
p
and is read as qth root of x
p
.
X (p/q) = (x 1/q ) p = (q√x) p and is read as pth power of qth root of x.
Solved Examples
1. Find the Value of (64) 2/3 ?
Solution:
= (4
3
)
2/3
= (4)
2
= 16
2. Find the value of (64/27)
5/3
?
Solution:
= (64/27)
5/3
= (4
3
/3
3
)
5/3
=((4/3)
3
)
5/3
= (4/3)
5
= 1024/243
3. Find the value of (256) 1/3 ?
Solution:
Given (256) 1/3
= (6 3 ) 1/3
= 6
Negative Rational Exponent
If x is a Non- Zero Rational Exponent and m is a positive integer then x -m = 1/x m = (1/x) m i.e. x -m is the reciprocal of x m .
The Same Rule is Applicable for Rational Exponents. Consider p/q to be a positive rational number and x > 0 is a rational number.
x -p/q = 1/x p/q = (1/x) p/q i.e. x -p/q is the reciprocal of x p/q
If x = a/b then (a/b) -p/q = (b/a) p/q
Solved Examples
1. Find 16 -1/2 ?
Solution:
Given 16 -1/2
= 1/16
1/2
=(1/16)
1/2
=((1/4)
2
)
1/2
= 1/4
2. Find the value of (32/243) -4/5 ?
Solution:
Given (32/243) -4/5
= 1/(32/243)
4/5
= (243/32)
4/5
= (3
5
/2
5
)
4/5
= ((3/2)
5
)
4/5
= (3/2)
4
= 81/16
