0580

Powers, Roots and Standard Form

Number

The required shape

  • Standard form writes a number as a × 10ⁿ, where n is an integer and a satisfies 1 ⩽ a < 10
  • The condition on a is what makes the form standard: exactly one non-zero digit sits before the decimal point
  • The sign of n records the size of the number
    • n is positive for numbers of 10 or more
    • n is zero for numbers from 1 up to but not including 10
    • n is negative for numbers between 0 and 1

Writing a number in standard form

  • Place the decimal point so that one non-zero digit lies in front of it, giving a
  • Count how many places the digits moved: that count is n
  • Moving the point left gives a positive index, moving it right gives a negative one
    • 68 500 becomes 6.85 × 10⁴
    • 0.00092 becomes 9.2 × 10⁻⁴
32 400 rewritten in standard form: four arrows above the digits each mark one multiplication by ten as the decimal point moves four places left from after the final zero to just after the 3, giving 3.24 × 10 to the power 4
Source: Converting to and from standard form by Save My Exams
0.0000324 rewritten in standard form: five arrows above the digits each mark one division by ten as the decimal point moves five places right to sit just after the 3, giving 3.24 × 10 to the power minus 5
Source: Converting to and from standard form by Save My Exams
Exam tip

The mark schemes treat "correct answer not in standard form" as a separate, lower-credit outcome, so 25 × 10⁻³ does not score what 2.5 × 10⁻² does. If asked why an expression is not standard form, name the broken condition: the first part must be at least 1 and below 10.

Worked example

Writing numbers in standard form

Write 62 700 000 and 0.000 47 in standard form.

Solution:

  • Standard form is a × 10ⁿ with 1 ⩽ a < 10
  • For 62 700 000, put the point after the first non-zero digit: 6.27
  • Count how far the point moved: from after the last zero to after the 6 is 7 places left
  • Moving left means a positive index, so 62 700 000 = 6.27 × 10⁷
  • For 0.000 47, the first non-zero digit is 4, giving 4.7
  • The point moved 4 places right, and moving right means a negative index
  • 0.000 47 = 4.7 × 10⁻⁴
  • Check each by counting back: 6.27 with the point moved 7 places right rebuilds 62 700 000