Rational Exponent | Positive, Negative Rational Exponents Definition & Examples

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:

= (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/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
= (3⁵/2⁵)^4/5
= ((3/2)⁵)^4/5
= (3/2)⁴
= 81/16