1 Exponents revisited

We have seen (in HELM booklet  1.2) the meaning to be assigned to the expression $a^p$ where $a$ is a positive number. We remind the reader that ‘ $a$ ’ is called the base and ‘ $p$ ’ is called the exponent (or power or index). There are various cases to consider:

If $m,n$ are positive integers

$\bullet$ $a^n=a\times a\times \⋯\times a$ with n terms

$\bullet$ $a^{1∕n}$ means the $n^{th}$ root of $a$ . That is, $a^{1∕n}$ is that positive number which satisfies

$\left(a^{1∕n}\right)\times \left(a^{1∕n}\right)\times \⋯\times \left(a^{1∕n}\right)=a$

where there are $n$ terms on the left hand side.

$\bullet$ $a^{m∕n}=\left(a^{1∕n}\right)\times \left(a^{1∕n}\right)\times \⋯\times \left(a^{1∕n}\right)$ where there are $m$ terms.

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

For convenience we again list the basic laws of exponents:

Example 1

Simplify the expression  $\frac{p^{n-2}{p}^m}{p^3{p}^{2m}}$

Solution

First we simplify the numerator:

$p^{n-2}{p}^m=p^{n-2+m}$

Next we simplify the denominator:

$p^3{p}^{2m}=p^{3+2m}$

Now we combine them and simplify:

$\frac{p^{n-2}{p}^m}{p^3{p}^{2m}}=\frac{p^{n-2+m}}{p^{3+2m}}=p^{n-2+m}{p}^{-3-2m}=p^{n-2+m-3-2m}=p^{n-m-5}$

Task!

Simplify the expression   $\frac{b^{m-n}{b}^3}{b^{2m}}$

First simplify the numerator:

Answer

Answer

Simplify the expression   $\frac{{\left(5a^m\right)}^2{a}^2}{{\left(a^3\right)}^2}$

Simplify the numerator:

Answer

${\left(5a^m\right)}^2{a}^2=25a^{2m}{a}^2=25a^{2m+2}$ Now include the denominator:

Answer

$\frac{{\left(5a^m\right)}^2{a}^2}{{\left(a^3\right)}^2}=\frac{25a^{2m+2}}{a^6}=25a^{2m+2-6}=25a^{2m-4}$

1.1 $a^x$ when $x$ is any real number

So far we have given the meaning of $a^p$ where $p$ is an integer or a rational number, that is, one which can be written as a quotient of integers. So, if $p$ is rational, then

$p=\frac{m}{n}\text{where}m,n\text{are integers}$

Now consider $x$ as a real number which cannot be written as a rational number. Two common examples of these irrational numbers are $\sqrt{2}$ and $\pi$ . What we shall do is approximate $x$ by a rational number by working to a fixed number of decimal places. For example if

$x=3.14159265\dots$

then, if we are working to 3 d.p. we would write

$x\approx 3.142$

and this number can certainly be expressed as a rational number:

$x\approx 3.142=\frac{3142}{1000}$

so, in this case

$a^x=a^{3.14159\dots }\approx a^{3.142}=a^{\frac{3142}{1000}}$

and the final term: $a^{\frac{3142}{1000}}$ can be determined in the usual way by calculator. Henceforth we shall therefore assume that the expression $a^x$ is defined for all positive values of $a$ and for $all$ real values of $x$ .

Task!

By working to 3 d.p. find, using your calculator, the value of $3^{\pi ∕2}$ .

First, approximate the value of $\frac{\pi }{2}$ :

Answer

$\frac{\pi }{2}\approx \frac{3.1415927\dots }{2}=1.5707963\⋯\approx 1.571$ Now determine $3^{\pi ∕2}$ :

Answer

$3^{\pi ∕2}\approx 3^{1.571}=5.618$ to 3 d.p.