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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:

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

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
hi
Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
how far
Abubakar
cool u
Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
Abegail Reply
hello
BenJay
hi
Method
I am eliacin, I need your help in maths
Rood
how can I help
Sir
hmm can we speak here?
Amoon
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
Arundhati Reply
When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
Jakoiya Reply
how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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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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