# 0.2 Essential mathematics

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## 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.

who are the alchemist?
alchemy science of transmutation. typically it is aim at tranforming lead to or other base metals to gold and the creation of the philosophers stone which in reality isn't a stone it's something priceless something we all need for coming times. don't be fooled
Kendrick
read Corinthians 5 verses 50 to the end of the chapter then read revelations chapter 2 verse 17
Kendrick
The word "Alchemy" comes from the forgotten name for Ancient Egypt, Khemmet. Khem was the name for the Egyptian Empire, but the actual land of Egypt was called Khemmet because the "T" on the end of a word denoted a physical location on Earth and not just an idea.
Michael
Wow!
mendie
What's the mass number of carbon
Akinbola
mass number of carbon is 12.
Nnenna
wat d atomic number of oxygen
safiya
atomic number of oxygen is 8
Nnenna
which quantum number divides shell into orbitals?
azimuthal
Emmanuel
hi
Charlie
azimuthal
reinhard
azimuthal
Charlie
what is atom
an atom is a smallest indivisible part of an element
Henry
an atom is the smallest part of an element that takes part in a chemical reaction
Nana
wat is neutralization
when any acid reacts with base to decrease it's acidity or vice-versa to form salt and solvent.. which is called neutralization
Santosh
explain buffer
Organic
buffer is a solution which resists changes in pH when acid or alkali added to it..
Santosh
hello, who is online
UTHMAN
buffer is the solution which resist the change in pH by addition of small amount of acid or alkali to it
KAUSIK
neutralisation is the process of mixing of a acid and a base to form water and corresponding salt
KAUSIK
how to solve equation on this
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Princewill
what is gases
Its one of the fundamental sate of matter alone side with liquid, solid and plasma
John
What is chemical bonding
John
To my own definitions. It's a unit of measurement to express the amount of a chemical substance.
What is mole
It's the unit of measurements used to express the amount of chemical substance.
Ozoaniehe
What is pressure
force over area
Jake
force applied per unit area
john
force applied per unit area
Prajapati
Why does carbonic acid don't react with metals
Why does carbonic acid don't react with metal
Some metals will react depending on their Standard Electrode Potential. Carbonic acid is a very weak acid (i.e. a low hydrogen ion concentration) so the rate of reaction is very low.
Paul
sample of carbon-12 has a mass of 6.00g. How many atoms of carbon-12 are in the sample
a sample of carbon-12 has a mass of 6.00g. How many atoms of carbon-12 are in the sample
an object of weight 10N immersed in a liquid displaces a quantity of d liquid.if d liquid displaced weights 6N.determine d up thrust of the object
how human discover earth is not flat
We don't fall off. If set off in any direction in a straight line and keep going. You'll end up back where you started.
earth is spherical
Unique
Also, every other planet is spherical as that is the most energy efficient shape. gravity pulls equally on all areas. Sphere.