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Standard Form

FoundationHigherAQAEdexcelOCR

Standard Form is a key number topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on writing and calculating with numbers in standard form, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. In a × 10ⁿ, the number a must be at least 1 but less than 10.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Standard form writes very large or very small numbers compactly as \(a \times 10^n\), where \(a\) is at least \(1\) and less than \(10\), and \(n\) is an integer.

The condition on \(a\) is strict and often overlooked. Writing \(24 \times 10^3\) is not standard form, because \(24\) is not between \(1\) and \(10\). It must be adjusted to \(2.4 \times 10^4\).

A positive power means a large number and a negative power means a small one. The index counts how many places the decimal point moves: to the right for positive, to the left for negative. Calculations follow the index laws, handling the numbers and the powers separately.

Revision notes

Writing numbers in standard form

Place the decimal point after the first non-zero digit, then count how many places it moved.

For \(4\,500\,000\), the point moves \(6\) places left, giving \(4.5 \times 10^6\). For \(0.00032\), it moves \(4\) places right, giving \(3.2 \times 10^{-4}\).

Multiplying and dividing

Multiply or divide the numbers, and add or subtract the indices using the index laws.

So \((3 \times 10^4) \times (2 \times 10^5) = 6 \times 10^9\). If the number part falls outside the range, adjust it: \(60 \times 10^8\) becomes \(6 \times 10^9\).

Adding and subtracting

The powers must match before you can add or subtract. Convert one number so both share the same index, then add the number parts.

For \(3 \times 10^5 + 4 \times 10^4\), rewrite as \(3 \times 10^5 + 0.4 \times 10^5 = 3.4 \times 10^5\).

Key points

  • Standard form is \(a \times 10^n\) with \(1 \le a < 10\).
  • A positive index means a large number.
  • A negative index means a number smaller than 1.
  • Multiply the numbers and add the indices.
  • Divide the numbers and subtract the indices.
  • Make the indices match before adding or subtracting.

Worked examples

Example 1

Write \(0.00047\) in standard form.

Working

\[4.7\]place the decimal point after the first non-zero digit
\[\text{point moves } 4 \text{ places right}\]the number is smaller than 1, so the index is negative
\[= 4.7 \times 10^{-4}\]write in standard form

Example 2

Work out \((5 \times 10^6) \times (4 \times 10^{-2})\).

Working

\[5 \times 4 = 20\]multiply the number parts
\[10^{6 + (-2)} = 10^4\]add the indices
\[20 \times 10^4 = 2 \times 10^5\]adjust so the number part is between 1 and 10

Example 3

Work out \((8 \times 10^7) \div (2 \times 10^3)\).

Working

\[8 \div 2 = 4\]divide the number parts
\[10^{7-3} = 10^4\]subtract the indices
\[= 4 \times 10^4\]combine the results

Common mistakes

  • Leaving the number part outside 1 to 10.

    20 × 10⁴ is not standard form. Adjust it to 2 × 10⁵.

  • Getting the sign of the index wrong.

    Numbers smaller than 1 take a negative index, so 0.0004 is 4 × 10⁻⁴, not 4 × 10⁴.

  • Adding the indices when adding the numbers.

    Index laws apply to multiplication and division only. To add, make the powers equal first.

  • Miscounting the decimal places.

    Count carefully from the original position to just after the first significant digit.

Exam tips

  • Always check the number part sits between 1 and 10 before writing your answer.
  • Handle the number parts and the powers as two separate steps.
  • Make the indices match before any addition or subtraction.
  • Use the standard form button on the calculator rather than typing out zeros.

Key terms

Standard form
A number written as \(a \times 10^n\) with \(1 \le a < 10\).
Index
The power of 10 showing the size of the number.
Integer
A whole number, positive, negative or zero.
Significant digit
A digit that contributes to the accuracy of a number.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.