Category: Powers and Exponents

Fractional (Rational) Exponents

Fractional exponent is a natural extension to the integer exponent.

We already know, that if bb is positive integer, then

  • ab=aaa...aba^b=\underbrace{a \cdot a \cdot a \cdot...\cdot a}_{b} (see positive exponent)
  • ab=1ab=1aaa...aba^{-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?

Adding Exponents

To understand addition of exponents, let's start from a simple example.

Example. Suppose, we want to find 2324{{2}}^{{3}}\cdot{{2}}^{{4}}.

We already learned about positive integer exponets, so we can write, that 23=222{{2}}^{{3}}={2}\cdot{2}\cdot{2} and 24=2222{{2}}^{{4}}={2}\cdot{2}\cdot{2}\cdot{2}.

Subtracting Exponents

To understand subtraction of exponents, let's start from a simple example.

Example. Suppose, we want to find 2724\frac{{{{2}}^{{7}}}}{{{{2}}^{{4}}}}.

We already learned about positive integer exponets, so we can write, that 27=2222222{{2}}^{{7}}={2}\cdot{2}\cdot{2}\cdot{2}\cdot{2}\cdot{2}\cdot{2} and 24=2222{{2}}^{{4}}={2}\cdot{2}\cdot{2}\cdot{2}.

Multiplying Exponents

To understand multiplication of exponents, let's start from a simple example.

Example. Suppose, we want to find (23)4{{\left({{2}}^{{3}}\right)}}^{{4}}.

We already learned about positive integer exponets, so we can rewrite outer exponent: (23)4=23232323{{\left({{2}}^{{3}}\right)}}^{{4}}={{2}}^{{3}}\cdot{{2}}^{{3}}\cdot{{2}}^{{3}}\cdot{{2}}^{{3}}.

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=am1n=(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}}}.

Properties of Exponents (Rules)

Properties (rules) of exponents:

  • Zero power: a0=1{{a}}^{{0}}={1}, a0{a}\ne{0}
  • Zero base: 0a=0{{0}}^{{a}}={0}, a0{a}\ne{0}
  • 00{{0}}^{{0}} is undefined
  • 1a=1{{1}}^{{a}}={1}
  • Negative exponent: ab=1ab{{a}}^{{-{b}}}=\frac{{1}}{{{a}}^{{b}}}, b0{b}\ne{0}
  • Nth root: a1n=an{{a}}^{{\frac{{1}}{{n}}}}={\sqrt[{{n}}]{{{a}}}}, n0{n}\ne{0}
  • Addition of exponents: aman=am+n{{a}}^{{m}}\cdot{{a}}^{{n}}={{a}}^{{{m}+{n}}}
  • Subtraction of exponents: aman=amn\frac{{{{a}}^{{m}}}}{{{{a}}^{{n}}}}={{a}}^{{{m}-{n}}}, a0{a}\ne{0}
  • Multiplication of exponents: (am)n=amn=(an)m{{\left({{a}}^{{m}}\right)}}^{{n}}={{a}}^{{{m}\cdot{n}}}={{\left({{a}}^{{n}}\right)}}^{{m}}
  • Division of exponents: amn=amn{\sqrt[{{n}}]{{{{a}}^{{m}}}}}={{a}}^{{\frac{{m}}{{n}}}}, n0{n}\ne{0}
  • amm=a{\sqrt[{{m}}]{{{{a}}^{{m}}}}}={a}, if m{m} is odd
  • amm=a{\sqrt[{{m}}]{{{{a}}^{{m}}}}}={\left|{a}\right|}, if m{m} is even
  • amn=(an)m{\sqrt[{{n}}]{{{{a}}^{{m}}}}}={{\left({\sqrt[{{n}}]{{{a}}}}\right)}}^{{m}} (just pay attention to signs and check, whether number exists)
  • Power of a product: anbn=(ab)n{{a}}^{{n}}\cdot{{b}}^{{n}}={{\left({a}{b}\right)}}^{{n}}
  • Power of a quotient: anbn=(ab)n\frac{{{{a}}^{{n}}}}{{{{b}}^{{n}}}}={{\left(\frac{{a}}{{b}}\right)}}^{{n}}, b0{b}\ne{0}

We already covered all rules earlier, except last two.