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