GCSE Maths Practice: standard-form

Question 3 of 10

A bacteria sample is measured at 3.5 × 10⁴ cells. Express this as an ordinary number.

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\( \begin{array}{l}\text{A bacteria sample contains }3.5 \times 10^4\text{ cells.}\\ \text{Write this in ordinary number form.}\end{array} \)

Choose one option:

For positive powers, move the decimal right. Each power adds one zero or one decimal shift to the right.

Converting from Standard Form to Ordinary Numbers

In GCSE Maths, it’s important to move comfortably between standard form and ordinary numbers. When a number is written in standard form with a positive power of ten, it represents a large number. To convert it, you multiply the first part (the coefficient) by ten for each value of the exponent. This moves the decimal point to the right.

Real-World Example

A colony of bacteria might be described as 3.5 × 10⁴ cells. That means 3.5 multiplied by 10⁴, or 3.5 × 10,000, which equals 35,000 cells. Writing this in standard form keeps it tidy and easier to read, especially when comparing with other values such as 4.2 × 10⁵ or 1.8 × 10⁶.

How to Convert Step by Step

  1. Identify the power of ten.
  2. Move the decimal point to the right if the power is positive (for large numbers).
  3. Move it to the left if the power is negative (for small numbers).
  4. Fill empty spaces with zeros as placeholders if needed.

Worked Example 1

Convert 6.2 × 10³ into ordinary form.

  • Move decimal three places right → 6200.
  • Result: 6,200.

Worked Example 2

Convert 2.08 × 10⁵ into ordinary form.

  • Move decimal five places right → 208000.
  • Answer: 208,000.

Worked Example 3

Convert 1.9 × 10² into ordinary form.

  • Move decimal two places right → 190.
  • Result: 190.

Common Mistakes

  • Moving the decimal in the wrong direction. Positive powers mean move right, negative powers mean move left.
  • Forgetting that each power adds one zero or one decimal shift.
  • Misplacing the decimal by one spot — always count carefully.

Applications in Science and Engineering

Standard form is common in biology, physics, and electronics. Scientists use it to record huge values like 3.5 × 10⁴ bacteria or small ones like 4.5 × 10⁻⁶ metres (cell diameters). It provides a universal system for expressing quantities at vastly different scales.

FAQs

  • Why move the decimal right? Because a positive exponent means multiplying by powers of ten, which increases the number’s size.
  • Do zeros after a whole number matter? Only if significant figures are required. 3.50 × 10⁴ means two significant figures.
  • Can standard form be used in everyday life? Yes — for example, when reading electricity bills, scientific data, or global distances.

Study Tip

When converting, visualise the power of ten as the number of steps your decimal moves. For example, 10⁴ = four steps to the right. Practise regularly by alternating between large and small numbers so you can confidently switch directions on exam questions.