Category: べき乗と指数

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.