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Simplify: x 8 x −3 .

x 11

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Simplify: y 8 y −6 .

y 13

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Convert from decimal notation to scientific notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and 0.004 . We know that 4,000 means 4 × 1,000 and 0.004 means 4 × 1 1,000 .

If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:

4,000 0.004 4 × 1,000 4 × 1 1,000 4 × 10 3 4 × 1 10 3 4 × 10 −3

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

Scientific notation

A number is expressed in scientific notation    when it is of the form

a × 10 n where 1 a < 10 and n is an integer

It is customary in scientific notation to use as the × multiplication sign, even though we avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation    .

This figure illustrates how to convert a number to scientific notation. It has two columns. In the first column is 4000 equals 4 times 10 to the third power. Below this, the equation is repeated, with an arrow demonstrating that the decimal point at the end of 4000 has moved three places to the left, so that 4000 becomes 4.000. The second column has 0.004 equals 4 times 10 to the negative third power. Below this, the equation is repeated, with an arrow demonstrating how the decimal point in 0.004 is moved three places to the right to produce 4.

In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.

The power of 10 is positive when the number is larger than 1: 4,000 = 4 × 10 3 The power of 10 is negative when the number is between 0 and 1: 0.004 = 4 × 10 −3

How to convert from decimal notation to scientific notation

Write in scientific notation: 37,000.

Solution

This figure is a table that has three columns and four 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. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.” The second cell reads “Remember, there is a decimal at the end of 37,000.” The third cell contains 37,000. One line down, the second cell reads “Move the decimal after the 3. 3.7000 is between 1 and 10.” In the second row, the first cell reads “Step 2. Count the number of decimal places, n, that the decimal place was moved. The second cell reads “The decimal point was moved 4 places to the left.” The third cell contains 370000 again, with an arrow showing the decimal point jumping places to the left from the end of the number until it ends up between the 3 and the 7. In the third row, the first cell reads “Step 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 to the n power. If it’s between 0 and 1, the power of 10 will be 10 to the negative n power.” The second cell reads “37,000 is greater than 1, so the power of 10 will have exponent 4.” The third cell contains 3.7 times 10 to the fourth power. In the fourth row, the first cell reads “Step 4. Check.” The second cell reads “Check to see if your answer makes sense.” The third cell reads “10 to the fourth power is 10,000 and 10,000 times 3.7 will be 37,000.” Below this is 37,000 equals 3.7 times 10 to the fourth power.
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Write in scientific notation: 96,000 .

9.6 × 10 4

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Write in scientific notation: 48,300 .

4.83 × 10 4

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Convert from decimal notation 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.

Write in scientific notation: 0.0052 .

Solution

The original number, 0.0052 , is between 0 and 1 so we will have a negative power of 10.

0.0052.
Move the decimal point to get 5.2, a number between 1 and 10. 0.0052, with an arrow showing the decimal point jumping three places to the right until it ends up between the 5 and 2.
Count the number of decimal places the point was moved. 3 places.
Write as a product with a power of 10. 5.2 times 10 to the power of negative 3.
Check.
5.2 × 10 −3 5.2 × 1 10 3 5.2 × 1 1000 5.2 × 0.001
0.0052 0.0052 equals 5.2 times 10 to the power of negative 3.
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Write in scientific notation: 0.0078 .

7.8 × 10 −3

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Write in scientific notation: 0.0129 .

1.29 × 10 −2

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Convert scientific notation to decimal form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

9.12 × 10 4 9.12 × 10 −4 9.12 × 10,000 9.12 × 0.0001 91,200 0.000912

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

9.12 × 10 4 = 91,200 9.12 × 10 −4 = 0.000912
This figure has two columns. In the left column is 9.12 times 10 to the fourth power equals 91,200. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the right. Because there are no digits after 2, the final two places are represented by blank spaces. Below this is the text “Move the decimal point four places to the right.” In the right column is 9.12 times 10 to the negative fourth power equals 0.000912. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the left. Because there are no digits before 9, the remaining three places are represented by spaces. Below this is the text “Move the decimal point 4 places to the left.”

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

Questions & Answers

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?
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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?
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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?
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5%
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12x
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-3/x
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-3x is correct
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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?
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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?
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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?
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Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
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26 + 37 = 63 + 8 = 71
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11+52=63+8=71
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23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
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23 +11 = 37. 23+37=63 63+8=71
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by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
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gayla
71-8-11 =52 is this correct?
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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
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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?
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DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
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Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
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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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