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8th Grade · 8.EE.A.3 · 8.EE.A.4

Scientific notation in 8th grade

Scientific notation is a short way to write very large and very small numbers, such as 3×1083 \times 10^8 for 300,000,000. A number in scientific notation is a number that is at least 1 and less than 10, multiplied by a power of 10. In 8th grade, your child uses scientific notation to estimate, compare sizes and work with numbers too long to write out.

What the standards say

The official Common Core wording, quoted exactly. The plain version follows.

8.EE.A.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

8.EE.A.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

What it means, in plain words

Estimating with powers of 10 8.EE.A.3

Powers of 10 are 10 multiplied by itself: 102=10010^2 = 100, 103=1,00010^3 = 1{,}000 and 106=1,000,00010^6 = 1{,}000{,}000. For these, the exponent counts the zeros. Negative exponents give small numbers: 10−1=0.110^{-1} = 0.1 and 10−3=0.00110^{-3} = 0.001.

The first skill is a rough estimate with a single digit in front. The United States has about 330 million people, which rounds to 3×1083 \times 10^8. The world has about 8 billion people, or 8×1098 \times 10^9. A tiny number works the same way: 0.00031 rounds to 3×10−43 \times 10^{-4}.

Children then use these estimates to say how many times bigger one quantity is. They divide the front numbers (the parts before the powers of 10) and subtract the exponents. For the two populations, 8×1093×108\frac{8 \times 10^9}{3 \times 10^8} is about 83×101\frac{8}{3} \times 10^1, which is about 27. So the world has more than 20 times as many people as the United States.

Your child should be able to:

  • Round a number to one digit and write it with a power of 10, such as 6,340,000≈6×1066{,}340{,}000 \approx 6 \times 10^6
  • Write a very small number the same way, such as 0.000072≈7×10−50.000072 \approx 7 \times 10^{-5}
  • Find how many times bigger one quantity is than another by dividing the front numbers and subtracting the exponents

Example

A city has about 9,000,000 people and a town has about 3,000 people. How many times as many people live in the city?

  1. Write each number with a power of 10: 9×1069 \times 10^6 and 3×1033 \times 10^3.
  2. Divide the front numbers: 9÷3=39 \div 3 = 3.
  3. Subtract the exponents: 106−3=10310^{6-3} = 10^3.
  4. 3×103=3,0003 \times 10^3 = 3{,}000, so the city has about 3,000 times as many people as the town.

Operations with scientific notation 8.EE.A.4

Operations with scientific notation use the exponent rules. Multiplying scientific notation means multiplying the front numbers and adding the exponents. Dividing means dividing the front numbers and subtracting the exponents. If the front number ends up 10 or more, or less than 1, the child moves the decimal point and changes the exponent to match.

Adding and subtracting take one more step, because the powers of 10 must match first. Problems often mix the two ways of writing a number. To add 6.2×1036.2 \times 10^3 and 4,500, the child writes 4,500 as 4.5×1034.5 \times 10^3. Then 6.2×103+4.5×103=10.7×1036.2 \times 10^3 + 4.5 \times 10^3 = 10.7 \times 10^3, which becomes 1.07×1041.07 \times 10^4.

Children also read scientific notation from a calculator and choose sensible units. A calculator shows 4.5×10−34.5 \times 10^{-3} as 4.5E-3, where E means "times 10 to the power". A good unit keeps the numbers readable. In some places the ocean floor spreads about 25 millimeters a year, which is easier to read than 0.000025 kilometers a year.

Your child should be able to:

  • Multiply and divide numbers in scientific notation and give the answer in scientific notation
  • Add and subtract after matching the powers of 10
  • Work with a mix of decimals and scientific notation in one problem
  • Read calculator answers such as 4.5E-3 and choose a sensible unit for a measurement

Example

Multiply (4×105)×(3×10−2)(4 \times 10^5) \times (3 \times 10^{-2}) and write the answer in scientific notation.

  1. Multiply the front numbers: 4×3=124 \times 3 = 12.
  2. Add the exponents: 5+(−2)=35 + (-2) = 3. The product is 12×10312 \times 10^3.
  3. 12 is not less than 10, so write 12=1.2×10112 = 1.2 \times 10^1 and add 1 to the exponent.
  4. The answer is 1.2×1041.2 \times 10^4, which is 12,000.

Where children usually get stuck

  • Your child rewrites 0.3 as 3×1013 \times 10^1.

    The child learned "move the decimal point and count the places" without the direction. A number smaller than 1 needs a negative exponent.

    What helps. Ask whether the new number is bigger or smaller than the old one. 3×1013 \times 10^1 is 30, far bigger than 0.3, so the exponent must be −1-1.

  • Your child multiplies (2×103)×(4×105)(2 \times 10^3) \times (4 \times 10^5) and writes 8×10158 \times 10^{15}.

    The front numbers were multiplied, so the exponents were multiplied too. The child copied the operation instead of using the exponent rule.

    What helps. Have your child write the powers as plain numbers once. 1,000 times 100,000 is 100,000,000, which is 10810^8. Counting the zeros shows that the exponents add.

  • Your child leaves an answer as 15×10415 \times 10^4 or 0.6×1030.6 \times 10^3.

    The arithmetic is right, but the front number must be at least 1 and less than 10. The child does not see this last step as part of the answer.

    What helps. Ask your child to read the front number aloud and check that it is at least 1 and less than 10. If it is not, move the decimal point one place and change the exponent by one: 15×104=1.5×10515 \times 10^4 = 1.5 \times 10^5.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

  1. 1Warm-up
    Match each scientific notation estimate to the decimal that rounds to it to 1 significant figure. One decimal will not be used.
    Connect each item on the left to its match on the right.
    Estimate
    Decimal
    3×10−4\displaystyle 3 \times 10^{-4}
    8×10−6\displaystyle 8 \times 10^{-6}
    0.000047\displaystyle 0.000047
    0.00031\displaystyle 0.00031
    0.0000082\displaystyle 0.0000082

    Use Tab to navigate between items. Click or tap a card to select it, then click or tap a card on the opposite side to create a match. You can also drag a card or its connector dot to a card on the opposite side.

    Show the answer and how to solve it

    Match 3×10−43 \times 10^{-4} with 0.00031 and 8×10−68 \times 10^{-6} with 0.0000082. The unused decimal is 0.000047 because it rounds to 5×10−55 \times 10^{-5}.

    1. 1Rewrite each decimal in scientific notation and round to 1 significant figure. Because the decimals span different exponents, counting zeros carefully identifies the correct match.
    2. 20.00031=3.1×10−40.00031 = 3.1 \times 10^{-4}, and 3.1 rounds to 3. 0.0000082=8.2×10−60.0000082 = 8.2 \times 10^{-6}, and 8.2 rounds to 8.
    3. 3The extra decimal is 0.000047. It is 4.7×10−54.7 \times 10^{-5}, which rounds to 5×10−55 \times 10^{-5}, so it does not match any estimate on the left.

    Common mistake. A common mistake is miscounting the zeros in 0.000047 and placing it in the 10−410^{-4} range instead of 10−510^{-5}. Rewriting as 4.7×10−54.7 \times 10^{-5} confirms it does not match any left-side estimate.

    The idea underneath. The exponent tells you the order of magnitude. Once you identify the correct power of 10 for each decimal, round the leading digit to find the one-digit coefficient.

  2. 2On level
    Select the list that is already ordered from least at the top to greatest at the bottom.
    A
    6.7×1048.4×1021.5×1023.2×10−1\displaystyle \begin{array}{l}6.7 \times 10^{4}\\8.4 \times 10^{2}\\1.5 \times 10^{2}\\3.2 \times 10^{-1}\end{array}
    B
    3.2×10−11.5×1028.4×1026.7×104\displaystyle \begin{array}{l}3.2 \times 10^{-1}\\1.5 \times 10^{2}\\8.4 \times 10^{2}\\6.7 \times 10^{4}\end{array}
    C
    1.5×1023.2×10−18.4×1026.7×104\displaystyle \begin{array}{l}1.5 \times 10^{2}\\3.2 \times 10^{-1}\\8.4 \times 10^{2}\\6.7 \times 10^{4}\end{array}
    D
    3.2×10−18.4×1021.5×1026.7×104\displaystyle \begin{array}{l}3.2 \times 10^{-1}\\8.4 \times 10^{2}\\1.5 \times 10^{2}\\6.7 \times 10^{4}\end{array}
    Show the answer and how to solve it

    The correct list is 3.2×10−13.2 \times 10^{-1}, 1.5×1021.5 \times 10^{2}, 8.4×1028.4 \times 10^{2}, 6.7×1046.7 \times 10^{4} from top to bottom because 10−1<102<10410^{-1} < 10^{2} < 10^{4} and, among the two 10210^{2} values, 1.5<8.41.5 < 8.4.

    1. 1All four choices show the same four values. The goal is to identify which list runs from least to greatest (top to bottom). Start by comparing exponents.
    2. 2Three distinct powers of 10 appear: 10−110^{-1}, 10210^{2}, and 10410^{4}. Since the coefficients are all between 1 and 10, a smaller exponent means a smaller value.
    3. 3Two values share the power 10210^{2}, so compare their coefficients: 1.5 and 8.4. Since 1.5<8.41.5 < 8.4, we get 1.5×102<8.4×1021.5 \times 10^{2} < 8.4 \times 10^{2}.
    4. 4Place all four values in one least-to-greatest chain, then select the list that matches this top-to-bottom order.

    Common mistake. A common mistake is swapping 1.5×1021.5 \times 10^{2} and 8.4×1028.4 \times 10^{2} by sorting only on the exponent. When exponents match, compare coefficients: 1.5<8.41.5 < 8.4. Also, 3.2×10−1=0.323.2 \times 10^{-1} = 0.32 is less than 1.5×102=1501.5 \times 10^{2} = 150, so it must come first.

    The idea underneath. For positive numbers in scientific notation, the power of 10 sets the size first; only when exponents match do you compare coefficients.

  3. 3On level
    Look at Tomas's estimate to one significant figure in scientific notation. Which statement best describes the work?
    6,340,000=6.34×106\displaystyle 6{,}340{,}000 = 6.34 \times 10^6
    ≈7×106\displaystyle \approx 7 \times 10^6
    A
    The leading digit should round down to 66, so the estimate should be 6×1066 \times 10^{6}.
    B
    The work is correct.
    C
    The leading digit should round down to 66, so the estimate should be 6×1076 \times 10^{7}.
    D
    6.34×1066.34 \times 10^{6} is already the estimate.
    Show the answer and how to solve it

    The best choice is the statement that the leading digit should round down to 66, so the estimate should be 6×1066 \times 10^{6}, because 6.346.34 rounds to 66 and the exponent stays 66.

    1. 1First check the scientific-notation step. Moving the decimal in 6,340,0006{,}340{,}000 six places left gives 6.34×1066.34 \times 10^{6}, so the first line is correct.
    2. 2To estimate with one significant figure, keep the 66 and look at the next digit. The next digit is 33, so 6.346.34 rounds down to 66.
    3. 3The exponent stays 66 because the rounded coefficient is 66, and 66 is between 11 and 1010.
    4. 4Tomas's last line shows 7×1067 \times 10^{6}, but the correct estimate is 6×1066 \times 10^{6}. The best statement is the one that says the leading digit should round down to 66.

    Common mistake. A common mistake is rounding up when the next digit is less than 55. Since the digit after 66 is 33, the leading digit rounds down to 66, not up to 77.

    The idea underneath. Estimating in scientific notation means rounding the coefficient to one significant figure; the exponent changes only if that rounding makes the coefficient become 10.

  4. 4Stretch
    Multiply the numbers below. Type the product in E-notation. For example, 2.7E6 means 2.7×1062.7 \times 10^{6}. The number before E must be at least 11 and less than 1010.
    (1.8×103)×(9.0×10−5)\displaystyle \left(1.8 \times 10^{3}\right) \times \left(9.0 \times 10^{-5}\right)
    Product in E-notation
    Show the answer and how to solve it

    The answer is 1.62E-1, which means 1.62×10−11.62 \times 10^{-1}.

    1. 1Look at the two numbers in scientific notation.
    2. 2Multiply the coefficients 1.81.8 and 9.09.0.
    3. 3Add the exponents 33 and −5-5 because both powers share base 1010.
    4. 4Combine the results. The raw product is 16.2×10−216.2 \times 10^{-2}. Since 16.216.2 is not between 11 and 1010, rewrite it as 1.62×1011.62 \times 10^{1} and increase the exponent by 11.
    5. 5The normalized product in E-notation is 1.62E-1.

    Common mistake. A common mistake is entering 16.2E-2. That is the raw product before normalizing. E-notation requires a coefficient at least 1 and less than 10, so rewrite 16.216.2 as 1.62×1011.62 \times 10^{1}, which shifts the exponent from −2-2 to −1-1.

    The idea underneath. When the coefficient becomes 10 or more, regroup one factor of 10 from the coefficient into the power of 10, which raises the exponent by 1 without changing the value.