# 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: $-{\sqrt[{{3}}]{{{2}}}}=-{{2}}^{{\frac{{1}}{{3}}}}$.