GCSEMathsAQAHigherRevision Notes1. NumberPowers, Roots & Standard FormOperations with Standard Form

Operations with Standard Form (AQA GCSE Maths): Revision Note

Exam code: 8300

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

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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 multiply and divide numbers in standard form without a calculator?

Multiplication

  • Consider the example 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

  • STEP 1
    Multiply the ordinary numbers

    • e.g. 3 cross times 4 equals 12

  • STEP 2
    Multiply the powers of 10 by adding the indices

    • e.g. 10 squared cross times 10 to the power of 5 equals 10 to the power of 2 plus 5 end exponent equals 10 to the power of 7

  • STEP 3
    Multiply the new ordinary number and the new power of 10

    • e.g. 12 cross times 10 to the power of 7

  • STEP 4 (if needed)
    Write the ordinary number in standard form and simplify the powers of 10 by adding the indices

    • e.g. 12 equals 1.2 cross times 10 to the power of 1 and 1.2 cross times 10 to the power of 1 cross times 10 to the power of 7 equals 1.2 cross times 10 to the power of 8

Division

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

  • STEP 1
    Divide the first ordinary number by the second ordinary number

    • e.g. 2 divided by 8 equals 0.25

  • STEP 2
    Divide the first power of 10 by the second power of 10 by subtracting the indices

    • e.g. 10 to the power of negative 5 end exponent divided by 10 to the power of negative 3 end exponent equals 10 to the power of negative 5 minus open parentheses negative 3 close parentheses end exponent equals 10 to the power of negative 2 end exponent

      • Be careful with negatives!

  • STEP 3
    Multiply the new ordinary number and the new power of 10

    • e.g. 0.25 cross times 10 to the power of negative 2 end exponent

  • STEP 4 (if needed)
    Write the ordinary number in standard form and simplify the powers of 10 by adding the indices

    • e.g. 0.25 equals 2.5 cross times 10 to the power of negative 1 end exponent and 2.5 cross times 10 to the power of negative 1 end exponent cross times 10 to the power of negative 2 end exponent equals 2.5 cross times 10 to the power of negative 3 end exponent

How do I add and subtract numbers in standard form without a calculator?

Method 1

  • Consider the example open parentheses 3.2 cross times 10 cubed close parentheses plus open parentheses 2.1 cross times 10 squared close parentheses

  • STEP 1
    Convert both numbers to ordinary numbers

    • e.g. 3.2 cross times 10 cubed equals 3200 and 2.1 cross times 10 squared equals 210

  • STEP 2
    Add or subtract the ordinary numbers

    • e.g. 3200 plus 210 equals 3410

  • STEP 3 (if needed)
    Convert the answer to standard form

    • e.g. 3410 equals 3.41 cross times 10 cubed

Method 2

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

  • STEP 1
    Rewrite the number with the biggest power of 10 so that it has the same power of 10 as the number with the lowest power of 10

    • e.g. 4 cross times 10 to the power of 50 equals 4 cross times 10 squared cross times 10 to the power of 48 equals 400 cross times 10 to the power of 48

      • The ordinary gets bigger as the power of 10 gets smaller

  • STEP 2
    Collect like terms by adding or subtracting the ordinary numbers

    • e.g. open parentheses 400 cross times 10 to the power of 48 close parentheses minus open parentheses 2 cross times 10 to the power of 48 close parentheses equals 398 cross times 10 to the power of 48

      • Do not change the power of 10

  • STEP 3
    Write the ordinary number in standard form and simplify the powers of 10 by adding the indices

    • e.g. 398 equals 3.98 cross times 10 squared and 3.98 cross times 10 squared cross times 10 to the power of 48 equals 3.98 cross times 10 to the power of 50

  • This method works for negative powers too

    • e.g. 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

      • 8 cross times 10 to the power of negative 20 end exponent equals 8 cross times 10 to the power of 1 cross times 10 to the power of negative 21 end exponent equals 80 cross times 10 to the power of negative 21 end exponent

      • open parentheses 80 cross times 10 to the power of negative 21 end exponent close parentheses minus open parentheses 5 cross times 10 to the power of negative 21 end exponent close parentheses equals 75 cross times 10 to the power of negative 21 end exponent

      • 75 equals 7.5 cross times 10 to the power of 1 and 7.5 cross times 10 to the power of 1 cross times 10 to the power of negative 21 end exponent equals 10 to the power of negative 20 end exponent

Examiner Tips and Tricks

The second method is the most efficient when the powers of 10 have large positive or large negative indices.

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.

Answer:

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.

Answer:

Rewrite 2.8 cross times 10 to the power of negative 6 end exponent so that the power of 10 matches 9.7 cross times 10 to the power of negative 8 end exponent

2.8 cross times 10 to the power of negative 6 end exponent equals 280 cross times 10 to the power of negative 8 end exponent

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

open parentheses 280 cross times 10 to the power of negative 8 end exponent close parentheses plus open parentheses 9.7 cross times 10 to the power of negative 8 end exponent close parentheses equals 289.7 cross times 10 to the power of negative 8 end exponent

Write the number in standard form

2.897 cross times 10 squared cross times 10 to the power of negative 8 end exponent equals 2.897 cross times 10 to the power of 2 minus 8 end exponent

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 Content Creator

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

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.