Dividing Exponents
We already saw division of exponents two times:
- when discussed fractional exponents $$$\left(a^{\frac{m}{n}}=\sqrt[n]{a^m}\right)$$$
- when discussed multiplication of exponents (indeed, $$${{a}}^{{\frac{{m}}{{n}}}}={{a}}^{{{m}\cdot\frac{{1}}{{n}}}}={{\left({{a}}^{{m}}\right)}}^{{\frac{{1}}{{n}}}}={\sqrt[{{n}}]{{{{a}}^{{m}}}}}$$$).
Rule for dividing exponents: $$$\color{purple}{\sqrt[n]{a^m}=a^{\frac{m}{n}}}$$$.
Word of caution. It is not always possible to interchange an exponent and a nth root, i.e. $$${\sqrt[{{n}}]{{{{a}}^{{m}}}}}\ne{{\left({\sqrt[{{n}}]{{{a}}}}\right)}}^{{m}}$$$.
It is always possible for positive number, but not for negative.
Example.
$$${\sqrt[{{4}}]{{{{\left(-{5}\right)}}^{{2}}}}}={\sqrt[{{4}}]{{{25}}}}={\sqrt[{{4}}]{{{{5}}^{{2}}}}}={{5}}^{{\frac{{2}}{{4}}}}={{5}}^{{\frac{{1}}{{2}}}}=\sqrt{{{5}}}$$$,
but $$${{\left({\sqrt[{{4}}]{{-{5}}}}\right)}}^{{2}}$$$ even doesn't exists, because $$${\sqrt[{{4}}]{{-{5}}}}$$$ doesn't exist.
Next couple of examples just show a couple of common problems.
Example 2. Rewrite, using positive exponents: $$${\sqrt[{{5}}]{{{{2}}^{{-\frac{{1}}{{3}}}}}}}$$$.
Just apply above rule: $$${\sqrt[{{5}}]{{{{2}}^{{-\frac{{1}}{{3}}}}}}}={{2}}^{{\frac{{-\frac{{1}}{{3}}}}{{5}}}}={{2}}^{{-\frac{{1}}{{15}}}}$$$.
Now, just rewrite using positive exponent: $$${{2}}^{{-\frac{{1}}{{15}}}}=\frac{{1}}{{{2}}^{{\frac{{1}}{{15}}}}}$$$.
What if we have a couple of radicals?
Example 3. Simplify: $$${\sqrt[{{4}}]{{{\sqrt[{{3}}]{{\frac{{1}}{{25}}}}}}}}$$$.
We start form innermost number: $$${\sqrt[{{4}}]{{{\sqrt[{{3}}]{{{{5}}^{{-{2}}}}}}}}}={\sqrt[{{4}}]{{{{5}}^{{-\frac{{2}}{{3}}}}}}}={{5}}^{{\frac{{-\frac{{2}}{{3}}}}{{4}}}}={{5}}^{{-\frac{{1}}{{6}}}}=\frac{{1}}{{{5}}^{{\frac{{1}}{{6}}}}}=\frac{{1}}{{\sqrt[{{6}}]{{{5}}}}}$$$.
Let's see how interchanging works.
Example 4. Simplify: $$${{\left({\sqrt[{{3}}]{{{{\left(-{2}\right)}}^{{4}}}}}\right)}}^{{6}}$$$.
$$${{\sqrt[{{3}}]{{{{\left(-{2}\right)}}^{{4}}}}}}^{{6}}={{\left({\sqrt[{{3}}]{{{16}}}}\right)}}^{{6}}={{16}}^{{\frac{{6}}{{3}}}}={{16}}^{{2}}={256}$$$.
Now, it is time to exercise.
Exercise 1. Rewrite, using positive exponents: $$${\sqrt[{{6}}]{{{{2}}^{{3}}}}}$$$.
Answer: $$${{2}}^{{\frac{{1}}{{2}}}}=\sqrt{{{2}}}$$$.
Exercise 2. Find $$${\sqrt[{{4}}]{{{{\left(-{5}\right)}}^{{2}}}}}$$$ and $$${{\left({\sqrt[{{4}}]{{-{5}}}}\right)}}^{{2}}$$$.
Answer: $$${\sqrt[{{4}}]{{-{{5}}^{{2}}}}}=\sqrt{{{5}}}$$$ and $$${{\left({\sqrt[{{4}}]{{-{5}}}}\right)}}^{{2}}$$$ doesn't exist.
Exercise 3. Find $$${\sqrt[{{5}}]{{{{\left(-{2}\right)}}^{{3}}}}}$$$.
Answer: $$${{\left(-{2}\right)}}^{{\frac{{3}}{{5}}}}$$$.
Exercise 4. Rewrite, using positive exponents: $$${\sqrt[{{4}}]{{{\sqrt[{{5}}]{{{{2}}^{{7}}}}}}}}$$$.
Answer: $$${{2}}^{{\frac{{7}}{{20}}}}$$$.
Exercise 5. Rewrite, using positive exponents: $$${\sqrt[{{5}}]{{{\sqrt[{{3}}]{{-{{2}}^{{5}}}}}}}}$$$.
Answer: $$$-{\sqrt[{{3}}]{{{2}}}}=-{{2}}^{{\frac{{1}}{{3}}}}$$$.