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 xm = y then we can express it in the form of x= (y)1/m. However, we can write y1/m = m√y and is referred to as the mth root of y.
y1/3 = 3√y, y1/5 = 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) = (xp)1/q = q√xp and is read as qth root of xp.
X(p/q) = (x1/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:
= (43)2/3
= (4)2
= 16
2. Find the value of (64/27)5/3?
Solution:
= (64/27)5/3
= (43/33)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
= (63)1/3
= 6
Negative Rational Exponent
If x is a Non- Zero Rational Exponent and m is a positive integer then x-m = 1/xm = (1/x)m i.e. x-m is the reciprocal of xm.
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/xp/q = (1/x)p/q i.e. x-p/q is the reciprocal of xp/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
= 1/161/2
=(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
= (35/25)4/5
= ((3/2)5)4/5
= (3/2)4
= 81/16