0580

Powers, Roots and Standard Form

Number

Multiplying and dividing

  • Deal with the ordinary numbers and the powers of ten separately
  • Multiply or divide the ordinary numbers as usual
  • Add the indices when multiplying, and subtract them when dividing
  • Check the result is still in standard form, and correct it if not
    • (5 × 10⁴) × (3 × 10⁶) gives 15 × 10¹⁰, which is rewritten as 1.5 × 10¹¹
    • (8 × 10³) ÷ (2 × 10⁷) gives 4 × 10⁻⁴, which already qualifies
  • Subtracting a negative index is where sign errors appear, so write the subtraction out in full

Adding and subtracting

  • Powers of ten cannot be added, so the two numbers must first be written with the same index
  • Rewrite the smaller number using the larger number's index, then add or subtract the ordinary parts
  • Restore standard form at the end if the result has drifted outside the range for a
Exam tip

Standard form appears in 17 of the 43 papers, and the first number must be at least 1 and below 10. After multiplying or dividing, check that condition again: 34 × 10⁵ is a correct value in the wrong form, and it is the form that is marked.

Worked example

Adding two numbers in standard form

Work out (4.2 × 10⁵) + (3.5 × 10⁴), giving the answer in standard form.

Solution:

  • The indices differ, so rewrite the second number with an index of 5
  • 3.5 × 10⁴ = 0.35 × 10⁵
  • Add the ordinary parts: 4.2 + 0.35 = 4.55
  • This gives 4.55 × 10⁵
  • Since 4.55 lies between 1 and 10, the answer is already in standard form: 4.55 × 10⁵
Worked example

Dividing when the result needs rewriting

Work out (1.2 × 10⁻²) ÷ (4 × 10³), giving the answer in standard form.

Solution:

  • Divide the ordinary numbers: 1.2 ÷ 4 = 0.3
  • Subtract the indices: −2 − 3 = −5, giving 10⁻⁵
  • This gives 0.3 × 10⁻⁵, which is not standard form because 0.3 is below 1
  • Write 0.3 as 3 × 10⁻¹, so the result becomes 3 × 10⁻¹ × 10⁻⁵
  • Add the indices: 3 × 10⁻⁶