GCSE Maths Practice: standard-form

Question 8 of 10

Practise expressing very small decimals in standard form using negative powers of ten.

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\( \begin{array}{l}\text{Write } 0.00000053 \text{ in standard form.}\end{array} \)

Choose one option:

When converting small numbers, move the decimal to the right and count carefully. Each move adds 1 to the negative exponent.

Expressing Tiny Numbers in Standard Form

When numbers become extremely small, writing them in full can be confusing and time-consuming. In GCSE Maths, we use standard form (scientific notation) to express these numbers neatly. A number written in standard form has two parts: a coefficient between 1 and 10, and a power of ten that shows how many places the decimal point has moved. For very small numbers, this power is negative because the decimal shifts to the right to reach a value between 1 and 10.

Why Standard Form Matters

In science, engineering, and computing, you often encounter values that are either enormous or microscopic. Standard form helps compare and calculate them easily without counting strings of zeros. It’s used to describe distances in space, wavelengths of light, or voltages in circuits. Learning to write tiny decimals in standard form helps students build number fluency and scientific literacy.

Converting a Small Number into Standard Form

  1. Find the decimal point in the original number.
  2. Move it until the first part of the number is between 1 and 10.
  3. Count how many times you moved the decimal point.
  4. Because the number is smaller than one, the power of ten becomes negative.

Worked Example 1

Convert 0.0000008 into standard form.

  • Move the decimal seven places right → 8
  • Exponent = −7
  • Result: 8 × 10⁻⁷

Worked Example 2

Convert 0.0000057 into standard form.

  • Move the decimal six places right → 5.7
  • Exponent = −6
  • Result: 5.7 × 10⁻⁶

Worked Example 3

Convert 0.000042 into standard form.

  • Move the decimal five places right → 4.2
  • Exponent = −5
  • Answer: 4.2 × 10⁻⁵

Common Mistakes

  • Using the wrong sign for the exponent. If the number is smaller than one, the exponent must be negative.
  • Moving the decimal the wrong direction (to the left instead of to the right).
  • Not ensuring the first number lies between 1 and 10.
  • Writing extra zeros after converting.

Applications in Real Life

Microscopic measurements such as cell sizes, bacteria lengths, and nanometre scales all rely on negative powers of ten. For instance, the diameter of a red blood cell is around 8 × 10⁻⁶ metres. Chemists use similar notation for molecular distances. Even astronomers apply standard form when describing the brightness of faint stars. Understanding this form ensures clear communication across different scientific disciplines.

FAQs

  • Why do we use a negative power? A negative exponent means the number is less than one, created by dividing by powers of ten.
  • Can we use standard form for zero? No, zero cannot be written in standard form because there’s no valid exponent that represents it.
  • How can I check my conversion? Multiply your result by the power of ten to see if it returns the original decimal.

Study Tip

Before converting, quickly estimate whether your number is small or large — this tells you whether the exponent will be negative or positive. Practise daily with both forms until counting decimal places feels natural. Confidence with standard form will help you in both maths and science GCSE exams.