Standard Form (Edexcel GCSE Maths: Foundation)

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Naomi C

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Naomi C

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Converting To & From Standard Form

What is standard form and why is it used?

  • Standard form is a way of writing very large and very small numbers using powers of 10
  • This allows us to:
    • Write them more concisely
    • Compare them more easily
    • Perform calculations with them more easily

How do I write a number in standard form?

  • Numbers written in standard form are always written as:

a cross times 10 to the power of n

  • Where:
    • 1 less or equal than a less than 10 (a is between 1 and 10)
    • n greater than 0 (n is positive) for large numbers
    • n less than 0 (n is negative) for small numbers
  • To write a large number such as 3 240 000 in standard form
    • Identify the value of a
      • 3.24
    • Find how many times you must multiply 3.24 by 10, to make 3 240 000
      • Count how many places you need to move the decimal point
      • We need to multiply by 10 six times
    • 3 240 000 = 3.24 × 10 × 10 × 10 × 10 × 10 × 10 = 3.24 × 106
  • To write a small number such as 0.000567 in standard form
    • Identify the value of a
      • 5.67
    • Find how many times you must divide 5.67 by 10, to make 0.000567
      • Count how many places you need to move the decimal point
      • We need to divide by 10 four times
      • We are dividing rather than multiplying so the power will be negative
    • 0.000567 = 5.67 ÷ 10 ÷ 10 ÷ 10 ÷ 10 = 5.67 × 10-4

Examiner Tip

  • On some calculators, typing in a very large or very small number and pressing box enclose equals will convert it to standard form

Worked example

(a)
Without a calculator, write 0.007052 in standard form.
 

Standard form will be written as a × 10n where a is between 1 and 10
Find the value for 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   (3 times)

Therefore n = -3.

0.007052 = 7.052 × 10-3

 
(b)
Without a calculator, write 324 500 000 in standard form.
 

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

a = 3.245

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

324 500 000  = 3.245 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10   (8 times)

Therefore n = 8

324 500 000 = 3.245 × 108 

 

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
      • 4 × 5 = 20 = 2 × 101
      • 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 once more
      • 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
      • 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
    • Choose the larger power
      • 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 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
    • Raise the number part to the power
    • Multiply the power of 10 by the new power
      • (3 × 105)2
      • 32 × (105)2
      • 9 × 1010
  • 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 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)table row blank row blank end table
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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Naomi C

Author: Naomi C

Expertise: Maths

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.