# 0.2 Exponentials  (Page 2/2)

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$\begin{array}{cccc}\hfill {a}^{-n}& =& 1÷a÷...÷a\hfill & \left(\mathrm{n times}\right)\hfill \\ \hfill & =& \frac{1}{1×a×\cdots ×a}\hfill & \left(\mathrm{n times}\right)\hfill \\ \hfill & =& \frac{1}{{a}^{n}}\hfill & \end{array}$

This means that a minus sign in the exponent is just another way of showing that the whole exponential number is to be divided instead of multiplied.

For example,

$\begin{array}{ccc}\hfill {2}^{-7}& =& \frac{1}{2×2×2×2×2×2×2}\hfill \\ & =& \frac{1}{{2}^{7}}\hfill \end{array}$

This law is useful in helping us simplify fractions where there are exponents in both the denominator and the numerator. For example:

$\begin{array}{ccc}\frac{{a}^{-3}}{{a}^{4}}& =& \frac{1}{{a}^{3}{a}^{4}}\\ & =& \frac{1}{{a}^{7}}\end{array}$

## Application using exponential law 3: ${a}^{-n}=\frac{1}{{a}^{n}},a\ne 0$

1. ${2}^{-2}=\frac{1}{{2}^{2}}$
2. $\frac{{2}^{-2}}{{3}^{2}}$
3. ${\left(\frac{2}{3}\right)}^{-3}$
4. $\frac{m}{{n}^{-4}}$
5. $\frac{{a}^{-3}·{x}^{4}}{{a}^{5}·{x}^{-2}}$

## Exponential law 4: ${a}^{m}÷{a}^{n}={a}^{m-n}$

We already realised with law 3 that a minus sign is another way of saying that the exponential number is to be divided instead of multiplied. Law 4 is just a more general way of saying the same thing. We get this law by multiplying law 3 by ${a}^{m}$ on both sides and using law 2.

$\begin{array}{ccc}\hfill \frac{{a}^{m}}{{a}^{n}}& =& {a}^{m}{a}^{-n}\hfill \\ \hfill & =& {a}^{m-n}\hfill \end{array}$

For example,

$\begin{array}{ccc}\hfill {2}^{7}÷{2}^{3}& =& \frac{2×2×2×2×2×2×2}{2×2×2}\hfill \\ & =& 2×2×2×2\hfill \\ & =& {2}^{4}\hfill \\ & =& {2}^{7-3}\hfill \end{array}$

## Application using exponential law 4: ${a}^{m}÷{a}^{n}={a}^{m-n}$

1. $\frac{{a}^{6}}{{a}^{2}}={a}^{6-2}$
2. $\frac{{3}^{2}}{{3}^{6}}$
3. $\frac{32{a}^{2}}{4{a}^{8}}$
4. $\frac{{a}^{3x}}{{a}^{4}}$

## Exponential law 5: ${\left(ab\right)}^{n}={a}^{n}{b}^{n}$

The order in which two real numbers are multiplied together does not matter. Therefore,

$\begin{array}{cccc}\hfill {\left(ab\right)}^{n}& =& a×b×a×b×...×a×b\hfill & \left(\mathrm{n times}\right)\hfill \\ \hfill & =& a×a×...×a\hfill & \left(\mathrm{n times}\right)\hfill \\ \hfill & & \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}×b×b×...×b\hfill & \left(\mathrm{n times}\right)\hfill \\ \hfill & =& {a}^{n}{b}^{n}\hfill & \end{array}$

For example,

$\begin{array}{ccc}\hfill {\left(2·3\right)}^{4}& =& \left(2·3\right)×\left(2·3\right)×\left(2·3\right)×\left(2·3\right)\hfill \\ & =& \left(2×2×2×2\right)×\left(3×3×3×3\right)\hfill \\ & =& \left({2}^{4}\right)×\left({3}^{4}\right)\hfill \\ & =& {2}^{4}{3}^{4}\hfill \end{array}$

## Application using exponential law 5: ${\left(ab\right)}^{n}={a}^{n}{b}^{n}$

1. ${\left(2xy\right)}^{3}={2}^{3}{x}^{3}{y}^{3}$
2. ${\left(\frac{7a}{b}\right)}^{2}$
3. ${\left(5a\right)}^{3}$

## Exponential law 6: ${\left({a}^{m}\right)}^{n}={a}^{mn}$

We can find the exponential of an exponential of a number. An exponential of a number is just a real number. So, even though the sentence sounds complicated, it is just saying that you can find the exponential of a number and then take the exponential of that number. You just take the exponential twice, using the answer of the first exponential as the argument for the second one.

$\begin{array}{cccc}\hfill {\left({a}^{m}\right)}^{n}& =& {a}^{m}×{a}^{m}×...×{a}^{m}\hfill & \left(\mathrm{n times}\right)\hfill \\ \hfill & =& a×a×...×a\hfill & \left(\mathrm{m}×\mathrm{n times}\right)\hfill \\ \hfill & =& {a}^{mn}\hfill & \end{array}$

For example,

$\begin{array}{ccc}\hfill {\left({2}^{2}\right)}^{3}& =& \left({2}^{2}\right)×\left({2}^{2}\right)×\left({2}^{2}\right)\hfill \\ & =& \left(2×2\right)×\left(2×2\right)×\left(2×2\right)\hfill \\ & =& \left({2}^{6}\right)\hfill \\ & =& {2}^{\left(2×3\right)}\hfill \end{array}$

## Application using exponential law 6: ${\left({a}^{m}\right)}^{n}={a}^{mn}$

1. ${\left({x}^{3}\right)}^{4}$
2. ${\left[{\left({a}^{4}\right)}^{3}\right]}^{2}$
3. ${\left({3}^{n+3}\right)}^{2}$

Simplify: $\frac{{5}^{2x-1}·{9}^{x-2}}{{15}^{2x-3}}$

1. $\begin{array}{ccc}& =& \frac{{5}^{2x-1}·{\left({3}^{2}\right)}^{x-2}}{{\left(5.3\right)}^{2x-3}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\hfill \\ & =& \frac{{5}^{2x-1}·{3}^{2x-4}}{{5}^{2x-3}·{3}^{2x-3}}\hfill \end{array}$
2. $\begin{array}{ccc}& =& {5}^{2x-1-2x+3}·{3}^{2x-4-2x+3}\hfill \\ & =& {5}^{2}·{3}^{-1}\hfill \end{array}$
3. $=\frac{25}{3}$

## Investigation : exponential numbers

Match the answers to the questions, by filling in the correct answer into the Answer column. Possible answers are: $\frac{3}{2}$ , 1, $-1$ , $-\frac{1}{3}$ , 8. Answers may be repeated.

 Question Answer ${2}^{3}$ ${7}^{3-3}$ ${\left(\frac{2}{3}\right)}^{-1}$ ${8}^{7-6}$ ${\left(-3\right)}^{-1}$ ${\left(-1\right)}^{23}$

We will use all these laws in Equations and Inequalities to help us solve exponential equations.

The following video gives an example on using some of the concepts covered in this chapter.

## Summary

• Exponential notation means a number written like ${a}^{n}$ where $n$ is an integer and $a$ can be any real number.
• $a$ is called the base and $n$ is called the exponent or index .
• The ${n}^{\mathrm{th}}$ power of $a$ is defined as: ${a}^{n}=a×a×\cdots ×a\phantom{\rule{2.em}{0ex}}\left(\mathrm{n times}\right)$
• There are six laws of exponents:
• Exponential Law 1: ${a}^{0}=1$
• Exponential Law 2: ${a}^{m}×{a}^{n}={a}^{m+n}$
• Exponential Law 3: ${a}^{-n}=\frac{1}{{a}^{n}},\phantom{\rule{1.em}{0ex}}a\ne 0$
• Exponential Law 4: ${a}^{m}÷{a}^{n}={a}^{m-n}$
• Exponential Law 5: ${\left(ab\right)}^{n}={a}^{n}{b}^{n}$
• Exponential Law 6: ${\left({a}^{m}\right)}^{n}={a}^{mn}$

## End of chapter exercises

1. Simplify as far as possible:
1. ${302}^{0}$
2. ${1}^{0}$
3. ${\left(xyz\right)}^{0}$
4. ${\left[{\left(3{x}^{4}{y}^{7}{z}^{12}\right)}^{5}{\left(-5{x}^{9}{y}^{3}{z}^{4}\right)}^{2}\right]}^{0}$
5. ${\left(2x\right)}^{3}$
6. ${\left(-2x\right)}^{3}$
7. ${\left(2x\right)}^{4}$
8. ${\left(-2x\right)}^{4}$
2. Simplify without using a calculator. Leave your answers with positive exponents.
1. $\frac{3{x}^{-3}}{{\left(3x\right)}^{2}}$
2. $5{x}^{0}+{8}^{-2}-{\left(\frac{1}{2}\right)}^{-2}·{1}^{x}$
3. $\frac{{5}^{b-3}}{{5}^{b+1}}$
3. Simplify, showing all steps:
1. $\frac{{2}^{a-2}.{3}^{a+3}}{{6}^{a}}$
2. $\frac{{a}^{2m+n+p}}{{a}^{m+n+p}·{a}^{m}}$
3. $\frac{{3}^{n}·{9}^{n-3}}{{27}^{n-1}}$
4. ${\left(\frac{2{x}^{2a}}{{y}^{-b}}\right)}^{3}$

5. $\frac{{2}^{3x-1}·{8}^{x+1}}{{4}^{2x-2}}$
6. $\frac{{6}^{2x}·{11}^{2x}}{{22}^{2x-1}·{3}^{2x}}$
4. Simplify, without using a calculator:
1. $\frac{{\left(-3\right)}^{-3}·{\left(-3\right)}^{2}}{{\left(-3\right)}^{-4}}$
2. ${\left({3}^{-1}+{2}^{-1}\right)}^{-1}$
3. $\frac{{9}^{n-1}·{27}^{3-2n}}{{81}^{2-n}}$
4. $\frac{{2}^{3n+2}·{8}^{n-3}}{{4}^{3n-2}}$

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