GCSE Maths Practice: standard-form

Question 2 of 10

Learn how to write small decimal numbers in standard form using negative powers of ten.

‹ ›

\( \begin{array}{l}\text{Write } 0.000035 \text{ in standard form.}\end{array} \)

Choose one option:

Move the decimal right to make the first number between 1 and 10. The number of moves becomes the negative exponent.

Understanding Standard Form for Small Numbers

In GCSE Maths, standard form is a way of writing very small or very large numbers more conveniently. When a number is smaller than one, it contains several zeros after the decimal point. Expressing it in standard form makes it easier to read, compare, and use in calculations. The key idea is to create a number between 1 and 10, then multiply it by a power of ten. If the number is less than one, that power is negative.

Why We Use It

Numbers such as 0.000035 can be hard to read at a glance, especially in scientific data or measurements. Standard form helps by summarising the scale of the number. It’s a skill not only useful for exams but also for fields such as physics, biology, and computing, where data values may vary by millions or billionths.

How to Convert to Standard Form

  1. Find where the decimal point currently is.
  2. Move it to make the number between 1 and 10.
  3. Count how many places you moved it.
  4. Since the number is smaller than one, use a negative power of ten equal to the number of moves.

Worked Example 1

Convert 0.0000048 into standard form.

  • Move the decimal six places right → 4.8
  • Exponent = −6
  • Answer: 4.8 × 10⁻⁶

Worked Example 2

Convert 0.00029 into standard form.

  • Move the decimal four places right → 2.9
  • Exponent = −4
  • Answer: 2.9 × 10⁻⁴

Worked Example 3

Convert 0.00000071 into standard form.

  • Move the decimal seven places right → 7.1
  • Exponent = −7
  • Answer: 7.1 × 10⁻⁷

Common Mistakes

  • Forgetting to make the number between 1 and 10 — for example, writing 35 × 10⁻⁶ instead of 3.5 × 10⁻⁵.
  • Confusing positive and negative exponents. Remember: smaller than one = negative power.
  • Counting the decimal places incorrectly. Double-check by multiplying your result back to confirm.

Real-World Applications

Standard form appears everywhere in science and technology. Chemists describe particle sizes, such as 4 × 10⁻⁸ metres for a molecule. Physicists use it to express forces and speeds. Computer scientists use it when working with floating-point numbers and precision data. Even in medicine, standard form helps in reporting concentrations or dosage values that are tiny fractions of a gram.

FAQs

  • Why does the exponent become negative? Each move to the right divides the number by ten, which reduces its value.
  • Can standard form start with a zero? No, the number before the × sign must be between 1 and 10.
  • What does the power tell us? It shows how many places the decimal has moved and whether the number is big or small.

Study Tip

When converting small decimals, always ask yourself: “How many tens would I need to multiply this by to make it between 1 and 10?” This will instantly help you determine the correct exponent. Practise with a range of values so you become fluent and avoid sign errors during your GCSE Maths exam.