Category: Powers and Exponents

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?

Adding Exponents

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

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

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

Subtracting Exponents

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

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

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

Multiplying Exponents

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

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

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

Dividing Exponents

We already saw division of exponents two times:

• when discussed fractional exponents $\left(a^{\frac{m}{n}}=\sqrt[n]{a^m}\right)$
• 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: $\color{purple}{\sqrt[n]{a^m}=a^{\frac{m}{n}}}$.

Properties of Exponents (Rules)

Properties (rules) of exponents:

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

We already covered all rules earlier, except last two.