We often have to multiply together the same number many times in math so we use the notation \[ b^n = \underbrace{bbb \cdots bb}_{n \text{ times} } \] to denote some number $b$ multiplied by itself $n$ times. In this section we will review the basic terminology associated with exponents and discuss their properties.

The fundamental ideas of exponents are:

• $b^n$: the number $b$ raised to the power $n$
  • $b$: the base
  • $n$: the exponent or power of $b$ in the expression $b^n$

By definitions, the zeroth power of any number is equal to one $b^0=1$.

We can also discuss exponential functions of the form $f:\mathbb{R} \to \mathbb{R}$ Define following functions:

• $b^x$: the exponential function base $b$
• $10^x$: the exponential function base $10$
• $\exp(x)=e^x$: the exponential function base $e$. The number $e$ is called Euler's number.
• $2^x$: the exponential function base $2$. This function is very important in computer science.

The number $e=2.7182818\ldots$ is a special base that has lots of applications. We call $e$ the natural base.

Another special base is $10$ because we use the decimal system for our numbers. We can write down very large numbers and very small numbers as powers of $10$. For example, one thousand can be written as $1\:000=10^3$, one million is $1\:000\:000=10^6$ and one billion is $1\:000\:000\:000=10^9$.

The following properties follow from the definition of exponentiation as repeated multiplication.

Multiplying together two exponential expressions with the same base is the same as adding the exponents: \[ b^m b^n = \underbrace{bbb \cdots bb}_{m \text{ times} } \underbrace{bbb \cdots bb}_{n \text{ times} } = \underbrace{bbbbbbb \cdots bb}_{m + n \text{ times} } = b^{m+n}. \]

Division by a number can be expressed as an exponent of minus one: \[ b^{-1} \equiv \frac{1}{b}. \] More generally any negative exponent corresponds to a division: \[ b^{-n} = \frac{1}{b^n}. \]

By combining Property 1 and Property 2 we obtain the following rule: \[ \frac{b^m}{b^n} = b^{m-n}. \]

In particular we have $b^{n}b^{-n}=b^{n-n}=b^0=1$. Multiplication by the number $b^{n}$ is the inverse operation of division by the number $b^{n}$. The net effect of the combination of both operations is the same as multiplying by one, i.e., the identity operation.

When an exponential expression is exponentiated, the inner exponent and the outer exponent multiply: \[ ({b^m})^n = \underbrace{(\underbrace{bbb \cdots bb}_{m \text{ times} }) (\underbrace{bbb \cdots bb}_{m \text{ times} }) \cdots (\underbrace{bbb \cdots bb}_{m \text{ times} })}_{n \text{ times} } = b^{mn}. \]

\[ (ab)^n =\underbrace{(ab)(ab)(ab) \cdots (ab)(ab)}_{n \text{ times} } = \underbrace{aaa \cdots aa}_{n \text{ times} } \underbrace{bbb \cdots bb}_{n \text{ times} } = a^n b^n. \]

\[ \left(\frac{a}{b}\right)^n = \underbrace{\left(\frac{a}{b}\right)\left(\frac{a}{b}\right)\left(\frac{a}{b}\right) \cdots \left(\frac{a}{b}\right)\left(\frac{a}{b}\right)}_{n \text{ times} } = \frac{ \overbrace{aaa \cdots aa}^{n \text{ times} } }{\underbrace{bbb \cdots bb}_{n \text{ times} } } = \frac{a^n}{b^n}. \]

Raising a number to the power $\frac{1}{n}$ is equivalent to finding the $n$th root of the number: \[ b^{\frac{1}{n}} = \sqrt[n]{b}. \] In particular, the square root corresponds to the exponent of one half $\sqrt{b}=b^{\frac{1}{2}}$. The cube root (the inverse of $x^3$) corresponds to $\sqrt[3]{b}\equiv b^{\frac{1}{3}}$. We can verify the inverse relationship between $\sqrt[3]{x}$ and $x^3$ using either Property 1: $(\sqrt[3]{x})^3=(x^{\frac{1}{3}})(x^{\frac{1}{3}})(x^{\frac{1}{3}})=x^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}=x^1=x$ or using Property 4: $(\sqrt[3]{x})^3=(x^{\frac{1}{3}})^3=x^{\frac{3}{3}}=x^1=x$.

Properties 5.1 and 5.2 also apply for fractional exponents: \[ \sqrt[n]{ab} = \sqrt[n]{a}\sqrt[n]{b}, \] \[ \sqrt[n]{\left(\frac{a}{b}\right)} = \frac{\sqrt[n]{a} }{ \sqrt[n]{b} }. \]

The function $f(x)=x^{n}$ behaves differently when the exponent $n$ is an even or odd. If $n$ is odd we have \[ \left( \sqrt[n]{b} \right)^n = \sqrt[n]{ b^n } = b. \]

However if $n$ is even the function $x^n$ destroys the sign of the number (e.g. $x^2$ which maps both $-x$ and $x$ to $x^2$). Thus the successive application of exponentiation by $n$ and the $n$th root has the same effect as the absolute value function: \[ \sqrt[n]{ b^n } = |b|. \] Recall that the absolute value function $|x|$ simply discards the information about the sign of $x$.

The expression $\left( \sqrt[n]{b} \right)^n$ cannot be computed whenever $b$ is a negative number. The reason is that we can't evaluate $\sqrt[n]{b}$ for $b<0$ in terms of real numbers (there is no real number which multiplied times itself an even number of times gives a negative number).

In science we often have to deal with very large numbers like the speed of light ($c=299\:792\:458$[m/s]), and very small numbers like the permeability of free space ($\mu_0=0.000001256637\ldots$[N/A$^2$]). It can be difficult to judge the magnitude of such numbers and to carry out calculations on them using the usual decimal notation.

Dealing with such numbers is much easier if we use scientific notation. For example the speed of light can be written as $c=2.99792458\times 10^{8}$[m/s] and the the permeability of free space is $\mu_0=1.256637\times 10^{-6}$[N/A$^2$]. In both cases we express the number as a decimal number between $1.0$ and $9.9999\ldots$ followed by the number $10$ raised to some power. The effect of multiplication by $10^8$ is to move the decimal point eight steps to the right thus making the number bigger. The effects of multiplying by $10^{-6}$ has the opposite effect of moving the decimal to the left thus making the number smaller. Scientific notation is very useful because it allows us to see clearly the size of numbers: $1.23\times 10^{6}$ is $1\:230\:000$ whereas $1.23\times 10^{-10}$ is $0.000\:000\:000\:123$. With scientific notation you don't have to count the zeros. Cool no?

The number of decimal places we use when specifying a certain physical quantity is usually an indicator of the precision with which we were able to measure this quantity. Taking into account the precision of the measurements we make is an important aspect of all quantitative research, but going into that right now would be a digression. If you want to read more about this, search for significant digits on the wikipedia page for scientific notation linked to below.

On computer systems, the floating point numbers are represented exactly like in scientific notation—a decimal part and an exponent. To separate the decimal part from exponent when entering a floating point number on the computer we use the character e, which stands for $\times 10^{?}$. For example to enter the permeability of free space into your calculator you should type 1.256637e-6.

http://en.wikipedia.org/wiki/Exponentiation

NOINDENT http://en.wikipedia.org/wiki/Scientific_notation