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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:

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
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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
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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
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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
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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
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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
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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
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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

The reaction of aceto nitrile with propane in the presence of the acid
Explain this paragraph in short
Manish Reply
What is solid state?
Manish Reply
What is chemical reaction
Manish
transforming reactants to product(s)
Andre
process
Andre
Example of Lewis acid
Chidera Reply
Example of Lewis acid
Chidera
Chlorine
Mikidad
Anything with an empty orbital... the hydrogen ion is the most common example. BH3 is the typical example, but any metal in a coordination complex can be considered a Lewis acid.
Eszter
okay thanks
Jovial
aluminium and sulphur react to give aluminium sulfide.How many grams of Al are required to produce 100g of aluminium sulphide
Soni Reply
aluminium and sulphur react to give aluminium sulphide how many grams of Al are required to produce 100g of aluminium sulphide?
Soni
aluminium and sulphur react to give aluminium sulphide how many grams of Al are required to produce 100g of aluminium sulphide?
Soni
2Al+3S=Al2S3
galina
m(Al)=100×27×2/150=36g
galina
150 comes from?
Soni
thank you very much
Soni
molar mass of Al2S3
galina
150.158
thiru
Why can't atom be created or destroyed
Jacaranda Reply
matter simply converts to pure energy
explain how to distinguish ethanol from a sample of ethanoic acid by chemical test
Alice Reply
explain how ethanol can be distinguished from ethanoic acid by chemical test
Alice
Using a suitable experiment, describe how diffusion occurs in gases.
Melody Reply
what is electrolytes?
charity Reply
substance which splits into ions during melting or dissolving
galina
on passing electric current though electrode
Kv
what is a radical
Jacob Reply
State that use law of partial pressure in a gas jar containing a gas and water what is the total pressure composed of 272cm^3 of carbon (iv) oxide were collected over water at15°c and 782mmHg pressure. calculate the volume of the dry gas at stp(SVP of water at 15°c is 12mmHg)
Aminat Reply
was Dalton's second postulate"atoms of the same kind have have similar/same mass and size" Or " the one mentioned in B here?
Maureen Reply
what was name of the Greek philosophers
Duku Reply
which of HF and HI is more acidic
Sahr Reply
HF is
HF
Duku
HI is stronger than HF (greater size of I courses greater length of bond)
galina
HI is a stronger acid due to less efficient orbital overlap. HF will react with with glass and extract calcium from bones, but those hazards are not because it's a stronger acid, but because it contains fluorine.
Eszter
hi
Victoria
hello
Jovial
hello
Asha
hi
Andre
hi
MO
hello
nanmya
hi
Hadiza
they are noble gases
Adum Reply

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Source:  OpenStax, Chemistry. OpenStax CNX. May 20, 2015 Download for free at http://legacy.cnx.org/content/col11760/1.9
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