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