One of the forms of algebraic expressions is power expression. It is in the form of a^{n}, where *a *is called as the base, and *n* is called as the exponent. The word exponent is so common that the power expressions are described as exponential expressions.

The value of the expression is the product of the base when multiplied by exponent number of times. That is, the product of *a* multiplied by *'n" *number of times. The exponent can be any real number. We will discuss here the case when the exponent is a fraction.

Related Calculators | |

Fractional Exponents Calculator | Exponent Fraction Calculator |

Fractions with Exponents Calculator | Negative Fractional Exponents Calculator |

The basic concept behind the fractional exponents are the laws of indices,

a^{m} x a^{n} = a^{m + n }and a^{m} ÷ a^{n} = a^{m – n }

If we use fractional exponents, the expression becomes

$a^{\frac{m}{n}} \times a^{\frac{m'}{n'}} = a^{\frac{m}{n} + \frac{m'}{n'}} = a^{\frac{mn' + m'n}{nn'}}$

and

$a^{\frac{m}{n}} \div a^{\frac{m'}{n'}} = a^{\frac{m}{n} - \frac{m'}{n'}} = a^{\frac{mn' - m'n}{nn'}}$

Let us take a simple case.

We know that, a is same as a^{1}, when expressed in exponential form.

Let the expression a^{n} represents the square root of *a* as a fractional exponent. Then by definition,

a^{n} x a^{n} = a^{1 } or a^{n + n } = a^{1} or a^{2n} = a^{1 }

Therefore, 2n = 1 or n = $\frac{1}{2}$

Hence the square of *a* = a^{1/2 }^{}

Generalizing the same concept, it can be established $\sqrt[n]{a}$ = a^{1/n}

The numerator of the fraction in a fractional exponent need not be 1 always and also it could be an improper fraction.

Let, a^{m/n} be a power expression with a fractional exponent.

Then as per law of indices, a^{m/n} = (a^{m})^{1/n}

= $\sqrt[n]{a^m}$, as per the concept established earlier.

The concept of fractional exponents greatly helps in simplifying radicals.

**Example 1:** Simplify 32^{1/4}

**Solution:** 32^{1/4 }= $\sqrt[4]{32}$= $\sqrt[4]{(2)(2)(2)(2)(2)}$ = 2^{5/4}

**Example 2:** Simplify 4^{3/2}

**Solution:** 4^{3/2} = $\sqrt{4^3}$ = $\sqrt{64}$ = 8