# 0.2 Essential mathematics

 Page 1 / 2

## Exponential arithmetic

Exponential notation is used to express very large and very small numbers as a product of two numbers. The first number of the product, the digit term , is usually a number not less than 1 and not greater than 10. The second number of the product, the exponential term , is written as 10 with an exponent. Some examples of exponential notation are:

$\begin{array}{ccc}\hfill 1000& =& 1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\hfill \\ \hfill 100& =& 1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2}\hfill \\ \hfill 10& =& 1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{1}\hfill \\ \hfill 1& =& 1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{0}\hfill \\ \hfill 0.1& =& 1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-1}\hfill \\ \hfill 0.001& =& 1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \\ \hfill 2386& =& 2.386\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1000=2.386\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\hfill \\ \hfill 0.123& =& 1.23\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}0.1=1.23\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-1}\hfill \end{array}$

The power (exponent) of 10 is equal to the number of places the decimal is shifted to give the digit number. The exponential method is particularly useful notation for every large and very small numbers. For example, 1,230,000,000 = 1.23 $×$ 10 9 , and 0.00000000036 = 3.6 $×$ 10 −10 .

Convert all numbers to the same power of 10, add the digit terms of the numbers, and if appropriate, convert the digit term back to a number between 1 and 10 by adjusting the exponential term.

Add 5.00 $×$ 10 −5 and 3.00 $×$ 10 −3 .

## Solution

$\begin{array}{ccc}\hfill 3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}& =& 300\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\hfill \\ \hfill \left(5.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\right)+\left(300\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\right)& =& 305\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}=3.05\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \end{array}$

## Subtraction of exponentials

Convert all numbers to the same power of 10, take the difference of the digit terms, and if appropriate, convert the digit term back to a number between 1 and 10 by adjusting the exponential term.

## Subtracting exponentials

Subtract 4.0 $×$ 10 −7 from 5.0 $×$ 10 −6 .

## Solution

$\begin{array}{}\\ 4.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}=0.40\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\\ \left(5.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\right)-\left(0.40\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\right)=4.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\end{array}$

## Multiplication of exponentials

Multiply the digit terms in the usual way and add the exponents of the exponential terms.

## Multiplying exponentials

Multiply 4.2 $×$ 10 −8 by 2.0 $×$ 10 3 .

## Solution

$\left(4.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(2.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\right)=\left(4.2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}2.0\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{\left(-8\right)+\left(+3\right)}=8.4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}$

## Division of exponentials

Divide the digit term of the numerator by the digit term of the denominator and subtract the exponents of the exponential terms.

## Dividing exponentials

Divide 3.6 $×$ 10 5 by 6.0 $×$ 10 −4 .

## Solution

$\frac{3.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}}{6.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}}\phantom{\rule{0.2em}{0ex}}=\left(\frac{3.6}{6.0}\right)\phantom{\rule{0.4em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{\left(-5\right)-\left(-4\right)}=0.60\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-1}=6.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}$

## Squaring of exponentials

Square the digit term in the usual way and multiply the exponent of the exponential term by 2.

## Squaring exponentials

Square the number 4.0 $×$ 10 −6 .

## Solution

${\left(4.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\right)}^{2}=4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}4\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(-6\right)}=16\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-12}=1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-11}$

## Cubing of exponentials

Cube the digit term in the usual way and multiply the exponent of the exponential term by 3.

## Cubing exponentials

Cube the number 2 $×$ 10 4 .

## Solution

${\left(2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}\right)}^{3}=2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}4}=8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{12}$

## Taking square roots of exponentials

If necessary, decrease or increase the exponential term so that the power of 10 is evenly divisible by 2. Extract the square root of the digit term and divide the exponential term by 2.

## Finding the square root of exponentials

Find the square root of 1.6 $×$ 10 −7 .

## Solution

$\begin{array}{ccc}\hfill 1.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}& =& 16\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}\hfill \\ \hfill \sqrt{16\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}}=\sqrt{16}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\sqrt{{10}^{-8}}& =\hfill & \sqrt{16}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-\phantom{\rule{0.2em}{0ex}}\frac{8}{2}}\phantom{\rule{0.2em}{0ex}}=4.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\hfill \end{array}$

## Significant figures

A beekeeper reports that he has 525,341 bees. The last three figures of the number are obviously inaccurate, for during the time the keeper was counting the bees, some of them died and others hatched; this makes it quite difficult to determine the exact number of bees. It would have been more accurate if the beekeeper had reported the number 525,000. In other words, the last three figures are not significant, except to set the position of the decimal point. Their exact values have no meaning useful in this situation. In reporting any information as numbers, use only as many significant figures as the accuracy of the measurement warrants.

what are ionic compound
where can I find the Arrhenius equation
ln(k two/k one)= (- activation energy/R)(1/T two - 1/T one) T is tempuratue in Kelvin, K is rate constant but you can also use rate two and one instead, R is 8.314 J/mol×Kelvin
Drenea
if ur solving for k two or k one you will need to use e to cancel out ln
Drenea
what is an atom
An atom is the smallest particle of an element which can take part in a chemical reaction..
olotu
hello
Karan
how to find the rate of reaction?
Karan
what is isomerism ?
Formula for equilibrium
is it equilibrium constant
olotu
Yes
Olatunde
yes
David
what us atomic of molecule
chemical formula for water
H20
Samson
what is elemental
what are the properties of pressure
Maryam
How can water be turned to gas
VICTOR
what's a periodic table
this can be defined as the systematic way of arranging element into vertical spaces called periods and horizontal spaces called groups on a table
onwuchekwa
how does carbon catenate?
condition in cracking from Diesel to petrol
hey I don't understand anything in chemistry so I was wondering if you could help me
i also
Okikiola
I also
Brient
hello
Brient
condition for cracking diesel to form kerosene
Brient
Really?
Isa
yes
Brient
can you tell me
Brient
Brient
what is periodic law
periodic law state that the physical and chemical properties of an element is the periodic function of their atomic number
rotimi
how is valency calculated
How is velency calculated
Bankz
Hi am Isaac, The number of electrons within the outer shell of the element determine its valency . To calculate the valency of an element(or molecule, for that matter), there are multiple methods. ... The valency of an atom is equal to the number of electrons in the outer shell if that number is fou
YAKUBU
what is the oxidation number of this compound fecl2,fecl3,fe2o3