## EXPONENTS

### AND THE FUNCTION

#Exponents and logarithms

Working with exponents is a skill that you have to master to survive IB Math, as it occurs in every second exercise. This onepager explains the main rules and tricks that you need to know, plus it presents the exponential function.

Explanations
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## PRACTICE [TASKS ACCORDING TO THIS SUBTOPIC]

#Exponents and logarithms

Simplify the following expression:

Other Onepagers in #Exponents and logarithms
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Laws of exponents

Exponents are a shortened way of representing the repeated multiplication of a number by itself.

For example the expression ${4}^{5}$ represents $4×4×4×4×4$, where 4 is the base number and 5 is the exponent. So the exponent counts the number of times we multiply the base to itself.

The exponent is also called a ‘power’ or an ‘index’.

Laws of exponents:

(1)  ${a}^{0}=1$                     (5)  $\frac{{a}^{n}}{{a}^{m}}={a}^{n-m}$                       (8)  ${a}^{\frac{1}{n}}=\sqrt[n]{a}$

(2)  ${a}^{1}=a$                     (6)  ${a}^{n}×{b}^{n}={\left(a×b\right)}^{n}$       (9)  ${a}^{\frac{n}{m}}=\sqrt[m]{{a}^{n}}$

(3)  ${a}^{-n}=\frac{1}{{a}^{n}}$               (7)  $\frac{{a}^{n}}{{b}^{n}}={\left(\frac{a}{b}\right)}^{n}$                    (10)  ${\left({a}^{n}\right)}^{m}={a}^{n×m}$

(4)  ${a}^{n}×{a}^{m}={a}^{n+m}$

#### Example #1

Simplify the following, leaving the answer in exponent form.

$\frac{\sqrt{a}×{a}^{\frac{2}{3}}}{\sqrt[3]{a}}⇒\left(8\right)⇒\frac{{a}^{\frac{1}{2}}×{a}^{\frac{2}{3}}}{{a}^{\frac{1}{3}}}⇒\left(4\right)⇒\frac{{a}^{\frac{1}{2}+\frac{2}{3}}}{{a}^{\frac{1}{3}}}$

$⇒\left(5\right)⇒{a}^{\frac{1}{2}+\frac{2}{3}-\frac{1}{3}}={a}^{\frac{4}{3}}$

#### Example #2

Simplify the following, leaving the answer in exponent form.

$\frac{{\left({x}^{2}{y}^{3}\right)}^{5}×{\left(x{y}^{2}\right)}^{3}}{{\left({x}^{2}\right)}^{2}×{\left(y{x}^{2}\right)}^{4}×{y}^{15}}⇒\left(6\right)&\left(10\right)⇒\frac{{x}^{2×5}×{y}^{3×5}×{x}^{3}×{y}^{2×3}}{{x}^{2×2}×{y}^{4}{x}^{2×4}×{y}^{15}}$

$⇒\left(5\right)⇒{x}^{13-12}=x$

#### Example #3

Simplify the following, leaving the answer in exponent form.

$\frac{{32}^{3}×{625}^{2}×{64}^{5}}{{128}^{4}×{25}^{6}}=\frac{{\left({2}^{5}\right)}^{3}×{\left({5}^{4}\right)}^{2}×{\left({2}^{6}\right)}^{5}}{{\left({2}^{7}\right)}^{4}×{\left({5}^{2}\right)}^{6}}⇒\left(10\right)⇒\frac{{2}^{15}×{5}^{8}×{2}^{30}}{{2}^{28}×{5}^{12}}$

$⇒\left(4\right)⇒\frac{{2}^{15+30}×{5}^{8}}{{2}^{28}×{5}^{12}}=\frac{{2}^{45}×{5}^{8}}{{2}^{28}×{5}^{12}}=\frac{{2}^{45}}{{2}^{28}}×\frac{{5}^{8}}{{5}^{12}}$

$⇒\left(5\right)⇒{2}^{45-28}×{5}^{8-12}={2}^{17}×{5}^{-4}=\frac{{2}^{17}}{{5}^{4}}$

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Exponential function

• An exponential function is a function of the form $f\left(x\right)={a}^{x}$, where a is $a$ positive real number, $a\ne 1$

• For large negative values of x, the y- value approaches the x-axis, but never reaches zero. So in this case the x-axis is a horizontal asymptote to the graph.

• The y-intercept is always (0,1), because ${a}^{0}=1$.

• If $a>1$, when x increases, so does y. This is called positive exponential, or exponential growth.

• If $0, when as x increases, y decreases. This is called a negative exponential, or exponential decay.
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Modelling with exponential functions

Many mathematical models of physical (half life of a radioactive substance), biological (population growth) or financial (compound interest) phenomena involve the concept of continuous growth or decay. A very useful application of the exponential function is that we can represent these forms of exponential growth or exponential decay.

We have a general formula to use:  $N=B{a}^{\frac{t}{k}}+c$

• when t=0 we have N=B+c, so B+c is the initial value of N, but you don’t have to memorise this, just remember that “initial” always means that you have to check the value of the function at t=0

• c is the background level (For example when we are modelling the cooling process of a liquid we know that the temperature of the liquid is not going under the temperature of the room, so the background level in this case is the temperature of the room. This is the value that the model approaches, i.e. N=c is the horizontal asymptote. Note that not every exercise is going to have a background level.)

• k is the time taken for the difference between N and the background level to increase by a factor of a, for example if a=2, k means the amount of time needed for B to doubles

• if a>1 then it’s a positive exponential, the function models exponential growth

• if 0<a<1 then it’s a negative exponential, the function models exponential decay

#### Example #1

A population of bacteria doubles its size in every three hours. The population growth can be modeled by a function N(t). Let’s assume that at the beginning we had a population of 100 bacteria. Build a function to model the growth of the population in the form: $N=B{a}^{\frac{t}{k}}$.

First let’s draw a table with values for t and our function N(t).

 t(in hours) N(t) 0 $100$ So when t=0, N(t)=100, because in the beginning we had 100 bacteria,that is our initial value 3 $100×2$ The population doubles its size in every three hours, so after three hours we are going to have 200 bacteria. 6 $100×{2}^{2}$ If we go another 3 hours, the population doubles again. 9 $100×{2}^{3}$ And so on... ...

We have an initial value of 100, and every three hours we multiply it with 2, so 2 is our common ratio. Let’s construct N(t):

$N\left(t\right)=100×{2}^{\frac{t}{3}}$

$\frac{t}{3}$ means how many three hour periods have passed by. If t=3, than 1 period, if t=6, than 2 periods, if t=4.5, than 1½ period has passed by, and so on.

It’s a simple example of modelling exponential growth.