Rational Exponents


Positive Exponents: If $ a$ is a real number and $ n$ is a natural number, the nth power of $ a$:

$\displaystyle a^n=a\cdot a \cdot \cdot \cdot a$

$ a$ is called the base, $ n$ is called the exponent and $ a^n$ is an exponential expression and is read as '$ a$ to the nth power' or '$ a$ to the nth'.

Negative Exponents: For all real numbers $ a \ne 0$, and all natural numbers $ n$, for which $ a^n$ is a real number

$\displaystyle a^{-n}=\frac{1}{a^n}$

$ n$th Root: If $ a$ is a real number, and $ n$ is a natural number for which $ a^{\frac{1}{n}}$

$ a^{\frac{1}{n}}$ is the $ n$th root of $ a$, that is $ (a^{\frac{1}{n}})^n=a$
if
either $ a\geq 0$ and $ n$ is an even positive integer
or $ n$ is odd positive integer

Rational Exponents: For all real number $ a$, and all positive integers $ m$ and $ n$ for which $ a^n$ is a real number

$\displaystyle a^{\frac{m}{n}}=(a^{\frac{1}{n}})^m$

Rules for Exponents For all integers $ m$ and $ n$ and all real numbers $ a$ and $ b$:

1. $ a^m a^n=a^{m+n}$
2. $ (a^{m})^n=a^{mn}$
3. $ (ab)^{m}=a^mb^m$
4. $ \Big(\frac{a}{b}\Big)^m=\frac{a^m}{b^m}$, $ b \ne 0$
5. $ \frac{1}{a^n}=a^{-n}$
6. $ \frac{a^m}{a^n}=a^{m-n}$, $ a \ne 0$
7. $ a^0=1$, $ a \ne 0$

Remark The above rules hold when $ m$ and $ n$ are rational numbers and $ a$ and $ b$ are positive real numbers.



Subsections


Department of Mathematics
Last modified: 2005-08-30