Fractional (Rational) Exponents

Fractional exponent is a natural extension to the integer exponent.

We already know, that if $b$ is positive integer, then

$a^b=\underbrace{a \cdot a \cdot a \cdot...\cdot a}_{b}$ (see positive exponent)
$a^{-b}=\frac{1}{a^b}=\frac{1}{\underbrace{a \cdot a \cdot a \cdot...\cdot a}_{b}}$ (see negative exponets)

But what if exponent is a fraction?

What will be the result of ${{a}}^{{\frac{{m}}{{n}}}}$ ?

We need nth root here:

$\color{purple}{a^{\frac{m}{n}}=\sqrt[n]{a^m}}$

Such numbers are called radicals (rational exponents).

Now, let's go through examples.

Example 1. Rewrite using exponent: $\sqrt{{{3}}}$ .

We can rewrite is ${\sqrt[{{2}}]{{{{3}}^{{1}}}}}$ .

Now, we clearly see, that $\sqrt{{{3}}}={{3}}^{{\frac{{1}}{{2}}}}$ .

Now, let's deal with negative exponents.

Example 2. Rewrite, using positive exponent: ${\sqrt[{{4}}]{{\frac{{1}}{{27}}}}}$ .

${\sqrt[{{4}}]{{\frac{{1}}{{27}}}}}={\sqrt[{{4}}]{{\frac{{1}}{{{3}}^{{3}}}}}}={\sqrt[{{4}}]{{{{3}}^{{-{3}}}}}}={{3}}^{{-\frac{{3}}{{4}}}}=\frac{{1}}{{{{3}}^{{\frac{{3}}{{4}}}}}}$ .

So, ${\sqrt[{{4}}]{{\frac{{1}}{{27}}}}}=\frac{{1}}{{{{3}}^{{\frac{{3}}{{4}}}}}}$ .

Now, let's do inverse operation.

Example 3. Rewrite, using radicals: ${{\left(\frac{{2}}{{5}}\right)}}^{{\frac{{3}}{{7}}}}$ .

We just go in another direction: ${{\left(\frac{{2}}{{5}}\right)}}^{{\frac{{3}}{{7}}}}={\sqrt[{{7}}]{{{{\left(\frac{{2}}{{5}}\right)}}^{{3}}}}}={\sqrt[{{7}}]{{\frac{{8}}{{125}}}}}$ .

Same applies to negative exponents.

Example 4. Rewrite, using radicals: ${{\left(\frac{{9}}{{5}}\right)}}^{{-\frac{{7}}{{8}}}}$ .

We have two ways here.

First is to get rid of minus on first step: ${{\left(\frac{{9}}{{5}}\right)}}^{{-\frac{{7}}{{8}}}}={{\left(\frac{{5}}{{9}}\right)}}^{{\frac{{7}}{{8}}}}={\sqrt[{{8}}]{{{{\left(\frac{{5}}{{9}}\right)}}^{{7}}}}}$ .

Second way is to get rid of minus at last: ${{\left(\frac{{9}}{{5}}\right)}}^{{-\frac{{7}}{{8}}}}={\sqrt[{{8}}]{{{{\left(\frac{{9}}{{5}}\right)}}^{{-{7}}}}}}={\sqrt[{{8}}]{{{{\left(\frac{{5}}{{9}}\right)}}^{{7}}}}}$ .

Now, exercise, to master this topic.

Exercise 1. Rewrite, using positive exponets: ${\sqrt[{{3}}]{{{5}}}}$ .

Answer: ${{5}}^{{\frac{{1}}{{3}}}}$ .

Exercise 2. Rewrite, using positive exponets: ${\sqrt[{{4}}]{{{{\left(\frac{{2}}{{3}}\right)}}^{{-{3}}}}}}$ .

Answer: ${{\left(\frac{{3}}{{2}}\right)}}^{{\frac{{3}}{{4}}}}$ .

Exercise 3. Rewrite, using radicals: ${{3}}^{{\frac{{2}}{{7}}}}$ .

Answer: ${\sqrt[{{7}}]{{{9}}}}$ .

Exercise 4. Rewrite, using radicals: ${{7}}^{{-\frac{{1}}{{2}}}}$ .

Answer: $\frac{{1}}{\sqrt{{{7}}}}$ .

Exercise 5. Rewrite, using radicals: ${{\left(\frac{{2}}{{5}}\right)}}^{{-\frac{{3}}{{7}}}}$ .

Answer: ${\sqrt[{{7}}]{{{{\left(\frac{{5}}{{2}}\right)}}^{{3}}}}}={\sqrt[{{7}}]{{\frac{{125}}{{8}}}}}$ .