GCSEMathsEdexcelRevision Notes1. NumberPowers, Roots & Standard FormStandard Form

Standard Form (Edexcel GCSE Maths)

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Converting To & From Standard Form

What is standard form?

  • Standard Form (sometimes called Standard Index Form) is a way of writing very big and very small numbers using powers of 10

Why do we use standard form?

  • Writing big (and small) numbers in Standard Form allows us to:
    • write them more neatly
    • compare them more easily
    • and it makes things easier when doing calculations

How do we use standard form?

  • Using Standard Form numbers are always written in the form:

a cross times 10 to the power of n

  • The rules:
    • 1 ≤ a < 10 so there is one non-zero digit before the decimal point
    • n > 0 for LARGE numbers – how many times a is multiplied by 10
    • n < 0 for SMALL numbers – how many times a is divided by 10
    • Do calculations on a calculator (if allowed)

Worked example

Without a calculator, write 0.007052 in standard form.

Standard form will be written as a × 10n. Ignore the place value and find the leading non-zero digit. Use this to find the value of a.

a = 7.052

The original number is smaller than 1 so n will be negative. Count how many times you need to divide a by 10 to get the original number.

0.007052 = 7.052 ÷ 10 ÷ 10 ÷ 10

Therefore n = -3.

0.007052 = 7.052 × 10-3 

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Operations with Standard Form

How do I multiply or divide two numbers in standard form?

  • If you can, use a calculator!
  • Otherwise multiply/divide the number parts first
    • If this answer is less than 1 or 10 or more then you will need to write it in standard form again
      • e.g. 4 × 5 = 20 = 2 × 101 or 2 ÷ 4 = 0.5 = 5 × 10-1
  • Then multiply/divide the powers of 10 using the laws of indices
  • Multiply the two parts together to get your answer in standard form
    • You might have to use the laws of indices one more
      • e.g. 4 × 102 × 5 × 107= 2 × 101 × 109 = 2 × 1010

How do I add or subtract two numbers in standard form?

  • If you can, use a calculator!
  • If the two numbers have the same power of 10 then you can simply add/subtract the number parts
    • If the answer is less than 1 or 10 or more then you will have to rewrite in standard form
      • e.g. 7 × 105 - 6.2 × 105 =  0.8 × 105 = 8 × 10-1 × 105 = 8 × 104 
  • Otherwise convert both numbers so that they have the same power of 10 (choosing the larger power)
    • e.g. 7 × 105 + 6 × 104 = 7 × 105 + 0.6 × 105 = 7.6 × 105
  • If the powers of 10 are small then you might find it easier to convert both numbers to ordinary numbers before adding/subtracting
    • You can convert your answer back to standard form if needed

How do I find powers or roots of a number in standard form?

  • If you can, use a calculator!
  • As standard form is two terms multiplied together you can split the power or root up
    • open parentheses a cross times 10 to the power of n close parentheses to the power of k equals a to the power of k cross times open parentheses 10 to the power of n close parentheses to the power of k equals a to the power of k cross times 10 to the power of n k end exponent
    • k-th root of a cross times 10 to the power of n end root equals k-th root of a cross times k-th root of 10 to the power of n end root equals k-th root of a cross times 10 to the power of n over k end exponent
  • Check to see whether you have to write your final answer in standard form

Worked example

(a)
Without using a calculator, multiply 5 cross times 10 to the power of 18 by 7 cross times 10 to the power of negative 4 end exponent.
Give your answer in standard form.

Separate into numbers and powers of 10.

table attributes columnalign right center left columnspacing 0px end attributes row cell 5 cross times 10 to the power of 18 cross times 7 cross times 10 to the power of negative 4 end exponent end cell equals cell 5 cross times 7 cross times 10 to the power of 18 cross times 10 to the power of negative 4 end exponent end cell end table

Multiply the integers together.
Use the laws of indices on the powers of 10.

equals 35 cross times 10 to the power of 18 plus open parentheses negative 4 close parentheses end exponent equals 35 cross times 10 to the power of 14

Adjust the first number, a, such that 1 less or equal than a less than 10.

equals 3.5 cross times 10 cross times 10 to the power of 14

Write in standard form.

bold 3 bold. bold 5 bold cross times bold 10 to the power of bold 15

(b)begin mathsize 16px style table row blank row blank end table end style
Use your calculator to find fraction numerator 1.275 cross times 10 to the power of 6 over denominator 3.4 cross times 10 to the power of negative 2 end exponent end fraction.
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.

Input the calculation into your calculator.
The result may or may not be in standard form.
Copy the digits, especially those zeros, carefully!

fraction numerator 1.275 cross times 10 to the power of 6 over denominator 3.4 cross times 10 to the power of negative 2 end exponent end fraction equals 37 space 500 space 000

Re-write in standard form.

bold 3 bold. bold 75 bold cross times bold 10 to the power of bold 7

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    Dan

    Author: Dan

    Expertise: Maths

    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.