IGCSEMathsCambridge (CIE)International MathsExtendedRevision NotesNumberPowers, Roots & Standard FormOperations with Standard Form

Operations with Standard Form (Cambridge (CIE) IGCSE International Maths)

Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Cambridge (CIE) International Maths: ExtendedEdexcel Maths A (Modular): Higher Unit 2Cambridge (CIE) International Maths: Core

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Operations with Standard Form

How do I perform calculations in standard form 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 standard form

    • 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 standard form, but the question requires it:

    • Either rewrite it using the standard process

      • e.g. 3 820 000 = 3.82 × 106

    • Or rewrite numbers in standard form, 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 standard form 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 standard form

      • 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 a 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 standard form, 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 standard form 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

Worked Example

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.

Write your answer in the form A cross times 10 to the power of n, where 1 less or equal than A less than 10 and n is an integer.

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 standard form, 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

Worked Example

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.

Write your answer in the form A cross times 10 to the power of n, where 1 less or equal than A less than 10 and n is an integer.

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 make the 9.7 a factor of 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
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths 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.