GCSE Maths Practice: standard-form

Question 7 of 10

The thickness of a sheet of metal is 0.0072 metres. Express this in standard form.

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\( \begin{array}{l}\text{The thickness of a metal sheet is }0.0072\text{ m. Write your answer in standard form.}\end{array} \)

Choose one option:

Always move the decimal to create a number between 1 and 10, then apply a negative exponent for small values.

Using Standard Form in Real Situations

Standard form isn’t just a classroom concept — it’s used every day in science, technology, and engineering. When numbers are very small, like the thickness of a paper sheet or the size of a cell, they are often expressed using negative powers of ten. This helps scientists and students compare measurements quickly without counting zeros.

Example Context

Suppose the thickness of a thin sheet of aluminium foil is 0.0072 metres. To write this more clearly, we can express it in standard form. We move the decimal three places to the right, giving 7.2 × 10⁻³ metres. This makes it easier to compare with other measurements, like 1.5 × 10⁻⁴ metres for a human hair strand’s thickness or 2 × 10⁻⁶ metres for a red blood cell.

How to Convert Step by Step

  1. Move the decimal so that the first number lies between 1 and 10.
  2. Count how many places it moved.
  3. Because the number is smaller than one, the power of ten is negative.

Worked Example 1

Convert 0.006 to standard form.

  • Move decimal three places right → 6
  • Exponent = −3
  • Answer: 6 × 10⁻³

Worked Example 2

Convert 0.00081 to standard form.

  • Move decimal four places right → 8.1
  • Exponent = −4
  • Result: 8.1 × 10⁻⁴

Worked Example 3

Convert 0.00000025 to standard form.

  • Move decimal seven places right → 2.5
  • Exponent = −7
  • Result: 2.5 × 10⁻⁷

Common Mistakes

  • Forgetting to use a negative exponent when the number is less than one.
  • Leaving the coefficient outside the range 1 ≤ a < 10.
  • Counting one place too many when moving the decimal.

Applications Beyond Maths

Standard form is heavily used in physics, biology, and even computing. In physics, speeds of light and sound are expressed using powers of ten. In biology, the diameters of cells and atoms use the same notation. In computing, memory or data transfer rates are often written using large powers of ten.

FAQs

  • Why is the exponent negative? Because the original number was less than one, and we had to move the decimal to the right.
  • Can we write 0.0072 × 10⁰? No — that would not be in standard form. Always adjust the power so that the first part is between 1 and 10.
  • What’s the quickest check? Multiply the number back out to confirm it gives 0.0072 again.

Study Tip

When you see a small number like 0.0072, quickly estimate how many zeros appear before the first non-zero digit. That count is the power for 10 with a negative sign. With practice, you can recognise common conversions instantly, such as 10⁻³ = 0.001.