Sometimes tested M2.8

Working with Standard Form

Standard form provides a concise way to write and manipulate very large or very small numbers. For the ESAT, you must be able to perform calculations (multiplication, division, addition, subtraction) with these numbers efficiently and without a calculator, paying close attention to the rules of indices.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • A number in standard form is written as a × 10n, where 'a' must be between 1 and 10 (i.e., 1 ≤ a < 10) and 'n' is an integer.
  • The power 'n' indicates the number's magnitude. A large positive 'n' means a large number; a negative 'n' means a number between 0 and 1.
  • When multiplying or dividing, handle the 'a' values and the powers of 10 separately. Use index laws: add powers for multiplication (10n × 10^m = 10^(n+m)) and subtract for division (10n / 10^m = 10^(n-m)).
  • When adding or subtracting, you must first adjust the numbers so they have a common power of 10. Then, you can add or subtract the 'a' values.
  • After any calculation, always check that your final answer is in correct standard form. This often requires an adjustment.
  • To order numbers in standard form, compare the powers of 10 ('n') first. If they are the same, then compare the 'a' values.

Formulae

(a × 10n) × (b × 10^m) = (a × b) × 10^(n+m)

Multiplying two numbers in standard form.

(a × 10n) / (b × 10^m) = (a / b) × 10^(n-m)

Dividing two numbers in standard form.

Definitions

Standard Form
A method of writing numbers as a product of a number between 1 and 10 (inclusive of 1) and a power of 10. For example, 5,200 is 5.2 × 103.

Worked example

Without a calculator, evaluate (1.5 × 108) / (3 × 10-4) and give your answer in standard form.

  1. 1

    Separate the calculation into two parts:

    the numbers and the powers of 10.

  2. 2

    Calculate the number part:

    1.5 / 3 = 0.5
  3. 3

    Calculate the power of 10 part using index laws for division:

    108 / 10-4 = 10^(8 - (-4)) = 1012.

  4. 4

    Combine the results:

    0.5 × 1012.

  5. 5

    This answer is not in standard form because 'a' (0.5) is not between 1 and 10.

  6. 6

    Convert 0.5 to standard form:

    0.5 = 5 × 10-1
  7. 7

    Substitute this back into the answer:

    (5 × 10-1) × 1012.

  8. 8

    Combine the powers of 10:

    5 × 10^(-1 + 12) = 5 × 1011.

Answer: 5 × 1011

Common mistakes

  • ×Forgetting to adjust the final answer into valid standard form. For example, leaving an answer as 45 × 105 instead of converting it to 4.5 × 106.
  • ×Making arithmetic errors with indices, particularly when subtracting a negative power during division, e.g., calculating 8 - (-4) as 4 instead of 12.
  • ×Incorrectly adding or subtracting numbers by just adding the 'a' parts without first making the powers of 10 equal.

No-calculator tips

  • ✓For division like 1.5 / 3, think of it as 15 / 3 = 5, then adjust for the decimal place. Since 1.5 is ten times smaller than 15, the answer is 0.5.
  • ✓When adding or subtracting, convert the number with the smaller power of 10 to match the larger one. For example, (4 × 106) + (5 × 105) becomes (4 × 106) + (0.5 × 106) = 4.5 × 106.
  • ✓To quickly check the magnitude of your answer, approximate the 'a' values to the nearest integer and perform the calculation with just the powers of 10.

Read this topic in the official UAT-UK ESAT guide →

All Mathematics 1 topics