Exponent Laws

Let $a$, $b$, $m$ and $n$ be any real numbers. Then the basic exponent rules are

• $a^{0} = 1$ ($a \neq 0$)
• $a^{-n} = \displaystyle \frac{1}{a^{n}}$ ($a \neq 0$)
• $a^{m}a^{n} = a^{m+n}$
• $\displaystyle \frac{a^{m}}{a^{n}} = a^{m-n}$ ($a \neq 0$)
• $(a^{m})^{n} = a^{mn}$

Distributive Laws and Rational Exponents

For a real number $a$ and a positive integer $n$, we use the $n$th-root notation. If $m$ is also an integer, then we have the following:

• $a^{1/n} = \sqrt[n]{a}\ $ (Note: these expressions are not defined when $n$ is even and $a<0$.)
• $a^{m/n} = \left( \sqrt[n]{a} \right)^{m} = \sqrt[n]{a^{m}}$.

Let $a$, $b$, $m$ and $n$ be any real numbers. The distributive laws of exponents are

• $\left( ab \right)^{m} = a^{m}b^{m}$
• $\displaystyle \left( \frac{a}{b} \right)^{m} = \frac{a^{m}}{b^{m}}$ ($b \neq 0$).

If we view rational exponents using $n$th-root notation, then we have the following as a consequence of the previous two properties:

• $\sqrt[n]{ab} = \sqrt[n]{a}\ \sqrt[n]{b}$   or   $(ab)^{1/n} = a^{1/n}b^{1/n}$
• $\displaystyle \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$   or   $\displaystyle \left( \frac{a}{b} \right)^{1/n} = \frac{a^{1/n}}{b^{1/n}}$   ($b \neq 0$).