6.7 Integer exponents and scientific notation  (Page 5/10)

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How to convert scientific notation to decimal form

Convert to decimal form: 6.2 × 10 3 .

Solution

This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads “Step 1. Determine the exponent, n, on the factor 10.” The second cell reads “The exponent is 3.” The third cell contains 6.2 times 10 cubed. In the second row, the first cell reads “Step 2. Move the decimal n places, adding zeros if needed. If the exponent is positive, move the decimal point n places to the right. If the exponent is negative, move the decimal point absolute value of n places to the left.” The second cell reads “The exponent is positive so move the decimal point 3 places to the right. We need to add two zeros as placeholders.” The third cell contains 6.200, with an arrow showing the decimal point jumping places to the right, from between the 6 and 2 to after the second 00 in 6.200. Below this is the number 6,200. In the third row, the first cell reads “Step 3. Check to see if your answer makes sense.” The second cell is blank. The third reads “10 cubed is 1000 and 1000 times 6.2 will be 6,200.” Beneath this is 6.2 times 10 cubed equals 6,200.
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Convert to decimal form: 1.3 × 10 3 .

1,300

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Convert to decimal form: 9.25 × 10 4 .

92,500

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The steps are summarized below.

Convert scientific notation to decimal form.

To convert scientific notation to decimal form:

  1. Determine the exponent, n , on the factor 10.
  2. Move the decimal n places, adding zeros if needed.
    • If the exponent is positive, move the decimal point n places to the right.
    • If the exponent is negative, move the decimal point | n | places to the left.
  3. Check.

Convert to decimal form: 8.9 × 10 −2 .

Solution

8.9 times 10 to the power of negative 2.
Determine the exponent, n , on the factor 10. The exponent is negative 2.
Since the exponent is negative, move the decimal point 2 places to the left. 8.9, with an arrow the decimal place showing the decimal point being moved two places to the left.
Add zeros as needed for placeholders. 8.9 times 10 to the power of negative 2 equals 0.089.
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Convert to decimal form: 1.2 × 10 −4 .

0.00012

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Convert to decimal form: 7.5 × 10 −2 .

0.075

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Multiply and divide using scientific notation

Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

Multiply. Write answers in decimal form: ( 4 × 10 5 ) ( 2 × 10 −7 ) .

Solution

( 4 × 10 5 ) ( 2 × 10 −7 ) Use the Commutative Property to rearrange the factors. 4 · 2 · 10 5 · 10 −7 Multiply. 8 × 10 −2 Change to decimal form by moving the decimal two places left. 0.08

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Multiply ( 3 × 10 6 ) ( 2 × 10 −8 ) . Write answers in decimal form.

0.06

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Multiply ( 3 × 10 −2 ) ( 3 × 10 −1 ) . Write answers in decimal form.

0.009

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Divide. Write answers in decimal form: 9 × 10 3 3 × 10 −2 .

Solution

9 × 10 3 3 × 10 −2 Separate the factors, rewriting as the product of two fractions. 9 3 × 10 3 10 −2 Divide. 3 × 10 5 Change to decimal form by moving the decimal five places right. 300,000

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Divide 8 × 10 4 2 × 10 −1 . Write answers in decimal form.

400,000

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Divide 8 × 10 2 4 × 10 −2 . Write answers in decimal form.

20,000

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Access these online resources for additional instruction and practice with integer exponents and scientific notation:

Key concepts

  • Property of Negative Exponents
    • If n is a positive integer and a ≠ 0 , then 1 a − n = a n
  • Quotient to a Negative Exponent
    • If a , b are real numbers, b ≠ 0 and n is an integer , then ( a b ) − n = ( b a ) n
  • To convert a decimal to scientific notation:
    1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
    2. Count the number of decimal places, n , that the decimal point was moved.
    3. Write the number as a product with a power of 10. If the original number is:
      • greater than 1, the power of 10 will be 10 n
      • between 0 and 1, the power of 10 will be 10 − n
    4. Check.

  • To convert scientific notation to decimal form:
    1. Determine the exponent, n , on the factor 10.
    2. Move the decimal n places, adding zeros if needed.
      • If the exponent is positive, move the decimal point n places to the right.
      • If the exponent is negative, move the decimal point | n | places to the left.
    3. Check.

Questions & Answers

The hypotenuse of a right triangle is 10cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle.
Edi Reply
Tickets for a show are $70 for adults and $50 for children. For one evening performance, a total of 300 tickets were sold and the receipts totaled $17,200. How many adult tickets and how many child tickets were sold?
Mum Reply
A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used?
Edi Reply
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold?
Jesus Reply
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Ronald Reply
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
Dojzae Reply
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
Xona Reply
5%
Michael
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Felecia answer 1.5 hours before he reaches her
Adriana Reply
I would like to solve the problem -6/2x
rachel Reply
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year?
Stephanie Reply
Tickets for the community fair cost $12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold?
Alpha Reply
220
gayla
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
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candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
26+37=63+8=71
ziad
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
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Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
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Toocute
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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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