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Dividing Exponents

We already saw division of exponents two times:

  • when discussed fractional exponents (amn=amn)\left(a^{\frac{m}{n}}=\sqrt[n]{a^m}\right)
  • when discussed multiplication of exponents (indeed, amn=am⋅1n=(am)1n=amn{{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: amn=amn\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. amn≠(an)m{\sqrt[{{n}}]{{{{a}}^{{m}}}}}\ne{{\left({\sqrt[{{n}}]{{{a}}}}\right)}}^{{m}}.

It is always possible for positive number, but not for negative.

Example.

(−5)24=254=524=524=512=5{\sqrt[{{4}}]{{{{\left(-{5}\right)}}^{{2}}}}}={\sqrt[{{4}}]{{{25}}}}={\sqrt[{{4}}]{{{{5}}^{{2}}}}}={{5}}^{{\frac{{2}}{{4}}}}={{5}}^{{\frac{{1}}{{2}}}}=\sqrt{{{5}}},

but (−54)2{{\left({\sqrt[{{4}}]{{-{5}}}}\right)}}^{{2}} even doesn't exists, because −54{\sqrt[{{4}}]{{-{5}}}} doesn't exist.

Next couple of examples just show a couple of common problems.

Example 2. Rewrite, using positive exponents: 2−135{\sqrt[{{5}}]{{{{2}}^{{-\frac{{1}}{{3}}}}}}}.

Just apply above rule: 2−135=2−135=2−115{\sqrt[{{5}}]{{{{2}}^{{-\frac{{1}}{{3}}}}}}}={{2}}^{{\frac{{-\frac{{1}}{{3}}}}{{5}}}}={{2}}^{{-\frac{{1}}{{15}}}}.

Now, just rewrite using positive exponent: 2−115=12115{{2}}^{{-\frac{{1}}{{15}}}}=\frac{{1}}{{{2}}^{{\frac{{1}}{{15}}}}}.

What if we have a couple of radicals?

Example 3. Simplify: 12534{\sqrt[{{4}}]{{{\sqrt[{{3}}]{{\frac{{1}}{{25}}}}}}}}.

We start form innermost number: 5−234=5−234=5−234=5−16=1516=156{\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: ((−2)43)6{{\left({\sqrt[{{3}}]{{{{\left(-{2}\right)}}^{{4}}}}}\right)}}^{{6}}.

(−2)436=(163)6=1663=162=256{{\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: 236{\sqrt[{{6}}]{{{{2}}^{{3}}}}}.

Answer: 212=2{{2}}^{{\frac{{1}}{{2}}}}=\sqrt{{{2}}}.

Exercise 2. Find (−5)24{\sqrt[{{4}}]{{{{\left(-{5}\right)}}^{{2}}}}} and (−54)2{{\left({\sqrt[{{4}}]{{-{5}}}}\right)}}^{{2}}.

Answer: −524=5{\sqrt[{{4}}]{{-{{5}}^{{2}}}}}=\sqrt{{{5}}} and (−54)2{{\left({\sqrt[{{4}}]{{-{5}}}}\right)}}^{{2}} doesn't exist.

Exercise 3. Find (−2)35{\sqrt[{{5}}]{{{{\left(-{2}\right)}}^{{3}}}}}.

Answer: (−2)35{{\left(-{2}\right)}}^{{\frac{{3}}{{5}}}}.

Exercise 4. Rewrite, using positive exponents: 2754{\sqrt[{{4}}]{{{\sqrt[{{5}}]{{{{2}}^{{7}}}}}}}}.

Answer: 2720{{2}}^{{\frac{{7}}{{20}}}}.

Exercise 5. Rewrite, using positive exponents: −2535{\sqrt[{{5}}]{{{\sqrt[{{3}}]{{-{{2}}^{{5}}}}}}}}.

Answer: −23=−213-{\sqrt[{{3}}]{{{2}}}}=-{{2}}^{{\frac{{1}}{{3}}}}.