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