<< Chapter < Page Chapter >> Page >

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:

1000 = 1 × 10 3 100 = 1 × 10 2 10 = 1 × 10 1 1 = 1 × 10 0 0.1 = 1 × 10 −1 0.001 = 1 × 10 −3 2386 = 2.386 × 1000 = 2.386 × 10 3 0.123 = 1.23 × 0.1 = 1.23 × 10 −1

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 .

Addition of exponentials

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.

Adding exponentials

Add 5.00 × 10 −5 and 3.00 × 10 −3 .

Solution

3.00 × 10 −3 = 300 × 10 −5 ( 5.00 × 10 −5 ) + ( 300 × 10 −5 ) = 305 × 10 −5 = 3.05 × 10 −3
Got questions? Get instant answers now!

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

4.0 × 10 −7 = 0.40 × 10 −6 ( 5.0 × 10 −6 ) ( 0.40 × 10 −6 ) = 4.6 × 10 −6
Got questions? Get instant answers now!

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

( 4.2 × 10 −8 ) × ( 2.0 × 10 3 ) = ( 4.2 × 2.0 ) × 10 ( −8 ) + ( +3 ) = 8.4 × 10 −5
Got questions? Get instant answers now!

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

3.6 × 10 −5 6.0 × 10 −4 = ( 3.6 6.0 ) × 10 ( −5 ) ( −4 ) = 0.60 × 10 −1 = 6.0 × 10 −2
Got questions? Get instant answers now!

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

( 4.0 × 10 −6 ) 2 = 4 × 4 × 10 2 × ( −6 ) = 16 × 10 −12 = 1.6 × 10 −11
Got questions? Get instant answers now!

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

( 2 × 10 4 ) 3 = 2 × 2 × 2 × 10 3 × 4 = 8 × 10 12
Got questions? Get instant answers now!

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

1.6 × 10 −7 = 16 × 10 −8 16 × 10 −8 = 16 × 10 −8 = 16 × 10 8 2 = 4.0 × 10 −4
Got questions? Get instant answers now!

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.

Questions & Answers

what is hydrocarbons compound?
Haile Reply
what is the 3d-orbital of Ti³+
Timi Reply
What is Lewis acids
Yabsra Reply
Lewis acid is any substance, such as the H+ ion, that can accept a pair of nonbonding electrons. In other words, a Lewis acid is an electron-pair acceptor. 
betuel
Thanks
Yabsra
🤜🤛
betuel
🤜🤛
betuel
describe the way of seperation of water and kerosene
Tang Reply
Kerosene is a hydrocarbon and non-polar. Water is a polar molecule. So a mixture of both liquids is immicible and by adding them to a separation funnel, you can open the tap flowing the less dense liquid in a container. You can read on bond polarity and separation techniques on Google.
Abdelkarim
kerosene will never with water cos its a immiscible liquid
SUNDAY
what is Chemistry
Papie Reply
Chemistry is a branch of natural light science
Yabsra
10 sentences discussing factors affecting solubility
Sara Reply
why is chemistry a science subject
Ukwumonu Reply
10 sentences discussing factors affecting solubility
Sara
How to name carbonique Atom
lix Reply
how many period do we have in the period table
Joseph Reply
187
Ukwumonu
how do i do ionic equations
Amantle Reply
what is the formula for alkanes
Amantle
CnH2n+2 is the alkane formula.
Walter
whenever you get off your fat arse
then you can start to do some real work
hmm
gabson
How much sodium hydroxide must be dissolved in 100mL of water to prepare a 3.95molL^_1
Cindy
what is vast array
benedict Reply
what is Nanoscience
benedict
from health care to manufacturing. Australian academy of science
what is the compound
Yaasmiin
what is Chemistry
Papie
What is array
Yabsra
what will be the total moles of all the molecule present when the different quantities of following gases are mixed together at step 4g of CH4, 22.4 dm3 of oxygen, 11.2dm3 of carbon dioxide and 3.02×10^23 molecules of ammonia.
Soni Reply
0.5 moles of methane and 0.5 mole of sulfur dioxide are mixed together what will be the mass of mixture. a.20g b.40g c.50g d.55g e.60g
Soni Reply
answer;b
Ravina
thnx
Soni
"the halogens are all oxidizing agents" what is the reason for this observation
Kelvin Reply
they are halogens....that is why numbnut 😁
This is not the place to talk down or be rude. Anonymous User keep your comments to yourself if you can't be more respectful.
Dionne
please what is a lightening thunder?
Onimisi Reply

Get the best Chemistry course in your pocket!





Source:  OpenStax, Chemistry. OpenStax CNX. May 20, 2015 Download for free at http://legacy.cnx.org/content/col11760/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Chemistry' conversation and receive update notifications?

Ask