In mathematics, exponent of a number tells how many times a number is multiplied by itself. Exponent can be whole number, integer or fractional.

An exponential expression in general would look as follows: b$^x$. Here, the ‘b’ is called the base and the x is called the exponent.
The exponent refers to the number of times the base is multiplied to itself.

For Example: 2$^3$=2*2*2
We see that since the exponent in the above example is 3, the 2 is multiplied to itself three times. Similarly if we have 3$^4$ that it would be equal to 3*3*3*3.

However these exponents are not restricted to positive integers only. The exponents can be negative and they can be fractions as well (proper or improper). In other words, the exponents can be rational numbers as well.

## What is Rational Exponent?

A rational number is a number of the form a/b where both a and b are integers and b is not equal to zero. Similarly a rational exponent therefore is an exponent of the form $\frac{a}{b}$.

In general an expression with a rational exponent would look as follows: $X$ $^\frac{a}{b}$

For Example:  $4$$^\frac{3}{2}, 10$$^\frac{-2}{3}$ etc. are all examples of expressions with rational exponents. Rational exponents can also be called fractional exponents. This is another method by which we can expression roots of bases or radicals of bases.

## Properties

The equations or expressions with rational exponents obey the following properties:
1) If an expression has a rational exponent $\frac{p}{q}$ where both p and q are positive integers, then that can be rewritten as qth root of pth power of the base. Symbolically that can be represented as shown in the following examples:

$4$ $^\frac{3}{2}$ = $\sqrt{4^3}$ = $(\sqrt{4})^3$

$100$ $^\frac{3}{5}$ = $\sqrt[5]{100^3}$= $(\sqrt[5]{100})^3$

In other words, the denominator of the rational exponent is in fact the index of the radical symbol, where as the numerator of the fractional exponent is the index (or the exponent) of the entire term or the radicand.

2) A negative rational exponent would mean that the expression is in the denominator or reciprocal. In general

$a^{(-m)}$ = $\frac{1}{a^m}$

For Example: $2$ $^{-\frac{1}{2}}$ = $\frac{1}{2^\frac{1}{2}}$ = $\frac{1}{\sqrt{2}}$

$7$ $^{-\frac{2}{5}}$ = $\frac{1}{2^{\frac{2}{5}}}$ = $\frac{1}{\sqrt[5]{7^2}}$

## Laws

Suppose s and t are real number bases and r and q are rational exponents then the following laws of exponents hold.
Note that these laws are same as those of whole number exponents:
1) $s^r \times s^q$ = $s^{(r+q)}$

2) $s^r \div s^q$ = $s^{r-q}$ ; s $\neq$ 0

3) $(s^r)^q$ = $s ^{rq}$

4) $(st)^r$ = $s^r * t^r$

5) $s^{-q}$ = $\frac{1}{s^q}$ ; s $\neq$ 0

6) $(\frac{s}{t})^q$ = $\frac{s^q}{t^q}$ ; t $\neq$ 0

## How to Solve?

The most popularly used strategy for solving rational exponents is to convert all the exponents to rational form from radical form and make all the exponents positive by bringing to numerator or denominator as required.
Then the exponents of same base can be added or subtracted or multiplied as the case may be.
Converting from the radical form to the rational expression form for solving problems sometimes makes the problem easier.
As we saw earlier also it is possible to convert rational exponential expression to a radical expression. Similarly it is also possible to convert a radical expression to rational exponential expression. The reverse process applies.

## Examples

This concept can be best understood with the help of some examples. Let us look at some rational exponents problems given below:
Example 1: Simplify the expression:

$\frac{(\sqrt[4]{4})* (\sqrt[4]{3^2})}{\sqrt[4]{6}}$

Solution: $\frac{(\sqrt[4]{4})* (\sqrt[4]{3^2})}{\sqrt[4]{6}}$

$4$ $^\frac{1}{4}$ * $\frac{3^\frac{2}{4}}{6^\frac{1}{4}}$

$(\frac{4}{6})^\frac{1}{4}$ * $3$ $^\frac{1}{2}$

$(\frac{2}{3})^\frac{1}{4}$ * $3$ $^\frac{1}{2}$

$\frac{2^\frac{1}{4} * 3^\frac{1}{2}}{3^\frac{1}{4}}$

$2$ $^\frac{1}{4}$ * $3$$^{\frac{1}{2} - \frac{1}{4}}$

$2$ $^\frac{1}{4}$ * $3$ $^{\frac{1}{4}}$

$(2*3)$ $^\frac{1}{4}$

$6$ $^\frac{1}{4}$ or $\sqrt[4]{6}$.

Example 2: Simplify

$(\sqrt{8}*\sqrt[8]{5})^2$

Solution: $(\sqrt{8}*\sqrt[8]{5})^2$

$(8^{\frac{1}{2}*2} * 5^{\frac{1}{8}*2})$

$8$ $^1*5{^\frac{1}{4}}$

$8(5{^\frac{1}{4}})$

Example 3: Simplify

$\frac{10}{10^\frac{1}{3}}$

Solution: $\frac{10}{10^\frac{1}{3}}$

$10$ $^{(1-\frac{1}{3})}$

$10$ $^{\frac{2}{3}}$

Example  4: Simplify

$\sqrt{49x^2 y^4}$

Solution: $\sqrt{49x^2 y^4}$

$(49x^2 y^4)^\frac{1}{2}$

$(7^2 x^2 y^4)^\frac{1}{2}$

$7^{2*\frac{1}{2}}x^{2*\frac{1}{2}}y^{4*\frac{1}{2}}$

$7^1 x^1 y^2$

$7xy^2$

Example  5: Simplify

$\sqrt[3]{-27a^3b^6}$

Solution: $\sqrt[3]{-27a^3b^6}$

$(-27a^3b^6)^\frac{1}{3}$

$((-3)^{3*\frac{1}{3}}a^{3*\frac{1}{3}}b^{6*\frac{1}{3}}$

$(-3)^1 a^1 b^2$

$-3ab^2$