GCSE Maths Practice: standard-form

Question 5 of 10

A liquid medicine contains 4.6 × 10⁻³ litres. Write this as an ordinary number.

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\( \begin{array}{l}\text{A medicine bottle contains }4.6 \times 10^{-3}\text{ litres.}\\ \text{Write this in ordinary form.}\end{array} \)

Choose one option:

For negative powers, divide by powers of ten or move the decimal left by the same number of places.

Converting Standard Form to Ordinary Numbers

Standard form is used in GCSE Maths to write numbers that are either very large or very small in a shorter way. When the power of ten is negative, the number is less than one and the decimal must be moved to the left. The number before the multiplication sign is called the coefficient, and the exponent tells us how far and in which direction the decimal must move.

Real-World Example

A droplet of medicine contains 4.6 × 10⁻³ litres of liquid. This is a very small amount, so it is written using a negative power of ten. To convert it into an ordinary number, we move the decimal point three places to the left, giving 0.0046 litres. This allows doctors and pharmacists to calculate safely without writing many zeros.

Step-by-Step Method

  1. Identify the exponent (here, −3).
  2. Because the exponent is negative, move the decimal to the left.
  3. Move it the same number of places as the exponent — three places.
  4. Fill any gaps with zeros.

Worked Example 1

Convert 3.2 × 10⁻².

  • Move decimal two places left → 0.032.
  • Answer: 0.032.

Worked Example 2

Convert 7.5 × 10⁻⁴.

  • Move decimal four places left → 0.00075.
  • Answer: 0.00075.

Worked Example 3

Convert 9.1 × 10⁻¹.

  • Move decimal one place left → 0.91.
  • Answer: 0.91.

Common Mistakes

  • Moving the decimal to the right instead of left when the exponent is negative.
  • Miscounting the number of decimal shifts, especially when zeros are involved.
  • Leaving out placeholder zeros instead of writing the full value.

Where This Appears in Real Life

Negative powers of ten are used for measurements in science and medicine, such as:

  • Dosage of chemicals and medicines
  • Micro-distances (micrometres, nanometres)
  • Electrical currents written in milliamps or microamps
  • Concentrations in chemistry, such as 2.5 × 10⁻⁵ mol

FAQs

  • Why does a negative power shrink the number? Because each step of 10⁻¹ divides the value by ten instead of multiplying it.
  • Can I check my answer? Yes — multiply 0.0046 back by 10³ to see if it returns to 4.6.
  • Does the coefficient have to stay between 1 and 10? Yes for standard form, but once converted to an ordinary number, that rule no longer applies.

Study Tip

A quick memory trick: negative exponent → number gets smaller → decimal moves left. Even if you forget the rule, imagine multiplying by 0.1 repeatedly — that helps you visualise why the number shrinks.