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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

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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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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