National 5MathsSQARevision NotesNumerical SkillsScientific NotationCalculations Using Scientific Notation

Calculations Using Scientific Notation (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

Calculations using scientific notation

How do I perform calculations in scientific notation using a calculator?

  • Make use of brackets around each number, and use the box enclose cross times 10 to the power of x end enclose button to enter numbers in scientific notation

    • e.g. open parentheses 3 cross times 10 to the power of 8 close parentheses cross times open parentheses 2 cross times 10 to the power of negative 3 end exponent close parentheses 

    • You can instead use the standard multiplication and index buttons

  • If your calculator answer is not in scientific notation, but the question requires it:

    • Either rewrite it using the standard process

      • e.g. 3 820 000 = 3.82 × 106

    • Or rewrite numbers in scientific notation, then apply the laws of indices

      • e.g.  243 × 1020 = (2.43 × 102) × 1020 = 2.43 × 1022

How do I perform calculations with numbers in scientific notation without a calculator?

Multiplication and division

  • Consider the "number parts" separately to the powers of 10

    • E.g. open parentheses 3 cross times 10 squared close parentheses space cross times space open parentheses 4 cross times 10 to the power of 5 close parentheses

      • Can be written as open parentheses 3 cross times 4 close parentheses cross times open parentheses 10 squared cross times 10 to the power of 5 close parentheses

    • Then calculate each part separately

      • Use laws of indices when combining the powers of 10

      • 12 cross times 10 to the power of 7

    • This can then be rewritten in scientific notation

      • 1.2 cross times 10 cross times 10 to the power of 7equals 1.2 cross times 10 to the power of 8

  • This process is the same for division

    • E.g. open parentheses 8 cross times 10 to the power of negative 5 end exponent close parentheses space divided by space open parentheses 2 cross times 10 to the power of negative 3 end exponent close parentheses

      • Can be written as fraction numerator 8 cross times 10 to the power of negative 5 end exponent over denominator 2 cross times 10 to the power of negative 3 end exponent end fraction equals 8 over 2 cross times 10 to the power of negative 5 end exponent over 10 to the power of negative 3 end exponent

    • Then calculate each part separately

      • Use laws of indices when combining the powers of 10

      • Be careful with negative powers -5 -(-3) is -5 + 3

      • 4 cross times 10 to the power of negative 2 end exponent

Addition and subtraction

  • One strategy is to write both numbers in full, rather than scientific notation, and then add or subtract them

    • E.g. open parentheses 3.2 cross times 10 cubed close parentheses plus open parentheses 2.1 cross times 10 squared close parentheses

    • Can be written as 3200 space plus thin space 210 space equals space 3410

    • Then this can be rewritten in scientific notation if needed, 3.41 cross times 10 cubed

  • However this method is not efficient for very large or very small powers

  • For very large or very small powers:

    • Write the values with the same, highest, power of 10

    • And then calculate the addition or subtraction, keeping the power of 10 the same

    • Consider open parentheses 4 cross times 10 to the power of 50 close parentheses plus open parentheses 2 cross times 10 to the power of 48 close parentheses

      • Rewrite both with the highest power of 10, i.e. 50

      • Changing 1048 to 1050 has made it 102 times larger, so make the 2 smaller by a factor of 102 to compensate

      • open parentheses 4 cross times 10 to the power of 50 close parentheses plus open parentheses 0.02 cross times 10 to the power of 50 close parentheses

      • These can now be added

      • 4.02 cross times 10 to the power of 50

    • Consider open parentheses 8 cross times 10 to the power of negative 20 end exponent close parentheses minus open parentheses 5 cross times 10 to the power of negative 21 end exponent close parentheses

      • Rewrite both with the higher power of 10, i.e. -20

      • Changing 10-21 to 10-20 has made it 101 times larger, so make the five 101 times smaller to compensate

      • open parentheses 8 cross times 10 to the power of negative 20 end exponent close parentheses minus open parentheses 0.5 cross times 10 to the power of negative 20 end exponent close parentheses

      • These can now be subtracted

      • 7.5 cross times 10 to the power of negative 20 end exponent

Examiner Tips and Tricks

Scientific notation questions will usually appear on Paper 2, on which you are allowed to use your calculator.

However it is still a good idea to know how to handle scientific notation calculations without a calculator.

Worked Example

A region of farmland has an area of 350 hectares.

There is an average of 2.42 cross times 10 to the power of 6 earthworms per hectare.

Calculate the number of earthworms in the region of farmland.

Give your answer in scientific notation.

Answer:

Use your calculator to multiply the number of hectares by the average number of earthworms per hectare

350 cross times open parentheses 2.42 cross times 10 to the power of 6 close parentheses equals 847 000 000 space

That answer from the calculator is not in scientific notation

Scientific notation will be written as a × 10n where a is between 1 and 10
Find the value for a

a equals 8.47

The original number is larger than 1 so n will be positive
Count how many times you need to multiply a by 10 to get the original number

space 847 000 000 equals 8.47 cross times 10 cross times 10 cross times 10 cross times 10 cross times 10 cross times 10 cross times 10 cross times 10
(8 times)

Therefore n = 8

8.47 cross times 10 to the power of 8

Worked Example

(a) Without using a calculator, find open parentheses 45 cross times 10 to the power of negative 3 end exponent close parentheses space divided by space open parentheses 0.9 cross times 10 to the power of 5 close parentheses. Give your answer in scientific notation.

(b) Without using a calculator, find open parentheses 2.8 cross times 10 to the power of negative 6 end exponent close parentheses space plus space open parentheses 9.7 cross times 10 to the power of negative 8 end exponent close parentheses. Give your answer in scientific notation.

Answer:

Part (a)

Rewrite the division as a fraction, then separate out the powers of 10

fraction numerator 45 cross times 10 to the power of negative 3 end exponent over denominator 0.9 cross times 10 to the power of 5 end fraction equals fraction numerator 45 over denominator 0.9 end fraction cross times 10 to the power of negative 3 end exponent over 10 to the power of 5

Work out fraction numerator 45 over denominator 0.9 end fraction

fraction numerator 45 over denominator 0.9 end fraction equals 450 over 9 equals 50

Work out 10 to the power of negative 3 end exponent over 10 to the power of 5 using laws of indices

10 to the power of negative 3 end exponent over 10 to the power of 5 equals 10 to the power of negative 3 minus 5 end exponent equals 10 to the power of negative 8 end exponent

Combine back together

open parentheses 45 cross times 10 to the power of negative 3 end exponent close parentheses space divided by space open parentheses 0.9 cross times 10 to the power of 5 close parentheses equals 50 cross times 10 to the power of negative 8 end exponent

Rewrite in scientific notation, where a is between 1 and 10

50 cross times 10 to the power of negative 8 end exponent equals 5 cross times 10 cross times 10 to the power of negative 8 end exponent equals 5 cross times 10 to the power of negative 7 end exponent

5 cross times 10 to the power of negative 7 end exponent

Part (b)

Rewrite both numbers with the highest power of ten, which is -6

Changing 10-8 to 10-6 has made it 102 times larger, so take 9.7 and make it 102 times smaller to compensate

open parentheses 2.8 cross times 10 to the power of negative 6 end exponent close parentheses plus open parentheses 0.097 cross times 10 to the power of negative 6 end exponent close parentheses

The numbers can now be added together, keeping the power of 10 the same

2.897 cross times 10 to the power of negative 6 end exponent

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.