Cambridge IGCSE Maths (0580) · Non-calculator · 1.8

Standard form without a calculator

Without a calculator you cannot type in very large or very small numbers, so you need to multiply and divide them in standard form using the index laws, and tidy the answer back into standard form.

Core and Extended
Paper 1 (Core) and Paper 2 (Extended)
7 questions
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The method

A number in standard form is \(A \times 10^n\), with \(1 \le A \lt 10\) and \(n\) an integer.
Multiplying: multiply the \(A\) parts and add the powers of 10.
Dividing: divide the \(A\) parts and subtract the powers of 10.
Adding or subtracting: write both numbers as ordinary numbers (or with the same power of 10) first.
If the \(A\) part ends up outside \(1 \le A \lt 10\), adjust it: \(24 \times 10^8 = 2.4 \times 10^9\) and \(0.4 \times 10^7 = 4 \times 10^6\).

Worked examples

Example 1 Core non-calculator · medium

Work out \((4 \times 10^5) \times (6 \times 10^3)\). Give your answer in standard form.

\(4 \times 6 = 24\) and \(10^5 \times 10^3 = 10^8\), so the product is \(24 \times 10^8\).
\(24 = 2.4 \times 10\), so \(24 \times 10^8 = 2.4 \times 10^9\).

Answer: \(2.4 \times 10^9\)

Example 2 Core non-calculator · medium

Work out \((3.2 \times 10^4) + (5 \times 10^3)\). Give your answer in standard form.

\(3.2 \times 10^4 = 32000\) and \(5 \times 10^3 = 5000\)
\(32000 + 5000 = 37000 = 3.7 \times 10^4\)

Answer: \(3.7 \times 10^4\)

Common slips

Leaving an answer such as \(24 \times 10^8\), which is not in standard form.
Adjusting the power the wrong way: making \(A\) smaller makes the power bigger.
Adding the powers when adding two numbers in standard form.
Negative powers: \(10^{-3}\) is \(0.001\), so \(5.2 \times 10^{-4} = 0.00052\).

Practice questions

No calculator. Write down every step, then open the worked answer.

Question 1 Core non-calculator · easy

Write \(0.00052\) in standard form.

Show the worked answer

Move the digits so that \(A = 5.2\): that is \(4\) places, and the number is less than \(1\), so the power is \(-4\).

Answer: \(5.2 \times 10^{-4}\)

Question 2 Core non-calculator · easy

Write \(6.07 \times 10^5\) as an ordinary number.

Show the worked answer

Multiply \(6.07\) by \(100000\): move the digits \(5\) places to the left.

Answer: \(607000\)

Question 3 Core non-calculator · medium

Work out \((2.5 \times 10^6) \times (4 \times 10^{-2})\). Give your answer in standard form.

Show the worked answer

\(2.5 \times 4 = 10\) and \(10^6 \times 10^{-2} = 10^4\), so the product is \(10 \times 10^4\).
\(10 \times 10^4 = 1 \times 10^5\)

Answer: \(1 \times 10^5\)

Question 4 Core non-calculator · medium

Work out \((9 \times 10^8) \div (3.6 \times 10^3)\). Give your answer in standard form.

Show the worked answer

\(9 \div 3.6 = 90 \div 36 = 2.5\)
\(10^8 \div 10^3 = 10^5\)

Answer: \(2.5 \times 10^5\)

Question 5 Core non-calculator · medium

Work out \((6.4 \times 10^5) - (8 \times 10^4)\). Give your answer in standard form.

Show the worked answer

\(640000 - 80000 = 560000\)
\(560000 = 5.6 \times 10^5\)

Answer: \(5.6 \times 10^5\)

Question 6 Core non-calculator · medium

Work out \((5 \times 10^3)^2\). Give your answer in standard form.

Show the worked answer

\((5 \times 10^3)^2 = 5^2 \times 10^6 = 25 \times 10^6\)
\(25 \times 10^6 = 2.5 \times 10^7\)

Answer: \(2.5 \times 10^7\)

Question 7 Core non-calculator · hard

A grain of sand has a mass of \(7 \times 10^{-4}\) grams.
How many grains of sand have a total mass of \(2.8\) kg? Give your answer in standard form.

Show the worked answer

\(2.8\) kg \(= 2800\) g \(= 2.8 \times 10^3\) g
\(\dfrac{2.8 \times 10^3}{7 \times 10^{-4}} = 0.4 \times 10^7\)
\(0.4 \times 10^7 = 4 \times 10^6\)

Answer: \(4 \times 10^6\)

Every example and question on this page is original, written for IGCSE Math Revision rather than taken from Cambridge papers, and each answer was re-solved independently before publishing. The syllabus is summarised in our own words: check the official 0580 syllabus.

Keep going

Standard Form questions for 0580: Core and Extended, each labelled calculator or non-calculator
Number practice for 0580, then My mistakes and a Skill Builder on what you missed
Standard form notes and the 0580 formula sheet
0580 past papers: time yourself on a specimen Paper 1 or Paper 2
All six non-calculator methods

Questions students ask

Which 0580 papers test standard form without a calculator?

Standard form is on both tiers (syllabus 1.8), so it can come up on Paper 1 (Core) and Paper 2 (Extended), the two non-calculator papers, as well as on the calculator papers.

Are these Cambridge questions?

No. The method is explained in our own words, and every example and question was written for IGCSE Math Revision and checked independently before publishing. For real papers, use the Cambridge copies linked from our 0580 past papers page.