Standard Form (Cambridge O Level Maths)

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:

• 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)

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
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• Check to see whether you have to write your final answer in standard form