Fractional exponents are a type of exponential notation that represents the **nth root of an expression** with an integer exponent.

The other names of Fractional exponents are”**Radicals**” or “**Rational Exponents**”

**For example,** the expression “a^{m/n}” represents the nth root of “a^{m}“.

In other words, we can say,

if “a^{m}” = x, then “a^{m/n} = x^{1/n}“.

Fractional exponents are useful for solving mathematical problems that include roots, such as finding the square root or cube root of a number.

## Fractional Exponents Rules

It’s important to understand the rules for simplifying expressions with fractional exponents, such as:

**Rule 1: **a^{1/m} × a^{1/n} = a^{(1/m + 1/n)}

**Rule 2:** a^{-m/n} = (1/a)^{m/n}

**Rule 3:** a^{1/m} ÷ a^{1/n} = a^{(1/m – 1/n)}

**Rule 4:** a^{1/m} ÷ b^{1/m} = (a÷b)^{1/m}

**Rule 5:** a^{1/m} × b^{1/m} = (ab)^{1/m}

With the help of these rules, you can solve expressions with fractional exponents and manipulate them to solve mathematical problems.

**Here are some examples of fractional exponents:**

Name of the exponent | Exponent | Indication |
---|---|---|

Square root | 1/2 | a^{1/2} = √a |

Cube root | 1/3 | a^{1/3} = ^{3}√a |

Fourth root | 1/4 | a^{1/4} = ^{4}√a |

### How to Divide Fractional Exponents?

To divide fractional exponents with the same base, you can subtract the exponent in the denominator from the exponent in the numerator.

In other words, if you have an expression of the form a^{m/n}/ a^{p/q}, where a is not equal to zero (base) and m, n, p, and q are integers, you can solve it as a^{(m/n – p/q)}.

**Note**: You may need to find a common denominator for the two fractions before performing the subtraction.

**For example:**

##### Solved (4(2/3)/ 4(1/2))

To divide fractional exponents with the same base, you can subtract the exponent in the denominator from the exponent in the numerator. In this case, we have a base of 4 and exponents of 2/3 and 1/2.

Now, we can rewrite the expression as:

(4(2/3)/ 4(1/2)) = 4^{(2/3 – 1/2)}

To subtract the exponents, we require to find a common denominator. The least common multiple of 3 and 2 is 6, so we can rewrite the exponents as:

2/3 – 1/2 = 4/6 – 3/6

= 1/6

Therefore, the solved expression is:

(4(2/3)/ 4(1/2)) = 4^{(1/6)}