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Fractional Exponents

One of the forms of algebraic expressions is power expression. It is in the form of an, 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.

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Fractional Exponents - Simple Fractions

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

am x an = am + n   and   am ÷ an = am – 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 a1, when expressed in exponential form.

Let the expression an represents the square root of a as a fractional exponent. Then by definition,

an x  an  = a1      or   an + n  =  a1       or    a2n  = a1

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

Hence the square of a =  a1/2

Generalizing the same concept, it can be established  $\sqrt[n]{a}$ = a1/n

Fractional Exponents - Harder Fractions

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

Let, am/n  be a power expression with a fractional exponent.

Then as per law of indices,  am/n  = (am)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  321/4

Solution:    321/4  =  $\sqrt[4]{32}$= $\sqrt[4]{(2)(2)(2)(2)(2)}$ = 25/4

Example 2:   Simplify  43/2

Solution:    43/2 =  $\sqrt{4^3}$ = $\sqrt{64}$ = 8

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