Exponents

If b is a complex number and n > 1 is an integer, the symbol

bⁿ

represents the product of n factors of b. Thus

$$\begin{array} {rcl} 3^3 & = & 27\ (1/2)^4 & = & 1/16\ (-3)^5 & = & -243\end{array}$$

The number b is called the base of bⁿ and the number n is called the exponent of the expression, or the power of b, or the logarithm of bⁿ in the base b if b > 0.

We if $b \neq 0$ we DEFINE

b⁰ = 1

as it would be nonsensical to speak of zero factors, and for any complex number, we DEFINE

b¹ = b

since it would be nonsensical to speak of the product of a single factor.

If n is a negative integer and $b \neq 0$ is a complex number we DEFINE

$$b^{n} = \frac{1}{b^{-n}} = \frac{1}{b^{\vert n\vert}}$$

since it would make no sense to speak of a product of a negative number of factors. For example,

$$(1+i)^{-1} = \frac{1}{1+i} = \frac{1}{2} - \frac{1}{2}i.$$

It is straightforward to check that if $a\neq 0$, $b \neq 0$ and m and n are any integers then

aⁿbⁿ = (ab)ⁿ,

$$b^n\times b^m = b^{m+n},$$

$$\frac{b^n}{b^m} = b^{n-m},$$

and

$$\left(b^n\right)^m = b^{nm}.$$

For example,

(2+i)³(2-i)³ = 5³,

$$(9+7i)^3\times (9+7i)^{-4} = (9+7i)^{-1} = \frac{1}{9+7i} = \frac{9}{130} -\frac{7}{130} i,$$

$$\frac{10^5}{10^{-2}} = 10^7$$

and

(2³)² = 64 = 2⁶.

We now restrict our attention to positive bases. If d is a positive integer and b > 0 we define

b^1/d

to be the POSITIVE solution to the equation x^d = b. For example,

8^1/3 = 2

since 2³ = 8 and 2 > 0.

Finally if n and d are integers and d > 0 then we define

$$b^{n/d} = \left(b^{1/d}\right)^n.$$

We have, then, for all b > 0 and all rational numbers r defined the symbol b^r. It can be checked in a straightforward matter that

a^rb^r = (ab)^r,

$$b^r\times b^s = b^{r+s},$$

$$\frac{b^r}{b^s} = b^{r-s}$$

and

$$\left(b^r\right)^s = b^{rs},$$

for any pair of rational numbers r and s, and any a > 0 and b > 0.

It should be noted that for some rational numbers r it is possible to extend the definition of b^r to negative bases. We shall not consider that here, and in the case of reciprocals of positive integers, that is 1/d, we suggest the notation $\sqrt[d]{b}$ to refer to solutions to x^d = b when $b \leq 0$ provided such solutions exist.

Exercises

1.

Write each expression in the form b^r:

(a)

6^1/36^3/5

(b)

(1/3)⁶(3/2)⁶

(c)

6^1/36^-3/5

(d)

$\left(6^{1/3}\right)^{-3/5}$

(e)

$${\frac{7^{-3/5}}{7^{-4/9}}}$$

(f)

3^1/35^1/3

2.

Combine exponents where possible.

(a)

(a²b³)⁵.

(b)

(a²b^-1)³(a² + b³).

(c)

$${\frac{(x^3y^2z^{-3})^2}{(xy^2z^3)^2}}$$.

(d)

$${\frac{(x^{1/3}y^2z^{-3})^2}{(xy^{-1/2}z^3)^2}}$$.

(e)

$${\frac{(x^{-1}y^2z^{-1/3})^5}{(x^{-1}y^2z^3)^{1/7}}}$$.