IGCSE · Practice questions · Standard form

Powers of ten, tamed.

Six original Cambridge-style questions on standard form: converting, ordering, calculating with and without a calculator, and giving answers in the correct form.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Front number between 1 and 10, then the power.

Shape: \(A \times 10^{n}\) with \(1 \le A < 10\) and \(n\) an integer.
Large or small: numbers of 10 or more have \(n > 0\); numbers below 1 have \(n < 0\).
Multiply/divide: front numbers together, then add/subtract the powers.
Add/subtract: match the powers first (or use ordinary numbers).
Tidy up: if the front number is 10 or more (or below 1), adjust it and the power.

Recall

[2 marks]

Write in standard form (a) \(820\,000\,000\) (b) \(0.000\,407\).

(a) \(8.2 \times 10^{8}\) ✓
(b) \(4.07 \times 10^{-4}\) ✓

Recall

[2 marks]

Write as ordinary numbers (a) \(5.03 \times 10^{5}\) (b) \(1.9 \times 10^{-2}\).

(a) \(503\,000\) ✓
(b) \(0.019\) ✓

Analysis

[2 marks]

Write these numbers in order of size, starting with the smallest.

\(4.5 \times 10^{-3}\) \(0.045\) \(4.5 \times 10^{-4}\) \(4.05 \times 10^{-3}\)

Write \(0.045 = 4.5 \times 10^{-2}\); the smallest power is \(-4\). ✓
Order: \(4.5 \times 10^{-4},\ 4.05 \times 10^{-3},\ 4.5 \times 10^{-3},\ 0.045\) ✓

Calculation

[2 marks]

Without a calculator, work out \((8 \times 10^{6}) \times (7 \times 10^{-2})\). Give your answer in standard form.

\(8 \times 7 = 56\) and \(10^{6} \times 10^{-2} = 10^{4}\), so \(56 \times 10^{4}\). ✓
\(56 = 5.6 \times 10^{1}\), so the answer is \(5.6 \times 10^{5}\). ✓

Calculation

[2 marks]

Without a calculator, work out \(3.2 \times 10^{4} + 9 \times 10^{3}\). Give your answer in standard form.

\(9 \times 10^{3} = 0.9 \times 10^{4}\). ✓
\(3.2 + 0.9 = 4.1\), so \(4.1 \times 10^{4}\). ✓

Analysis

[3 marks]

The distance from the Earth to the Sun is \(1.5 \times 10^{8}\) km. Light travels at \(3 \times 10^{5}\) km per second.

Calculate the time, in minutes, that light takes to travel from the Sun to the Earth.

Time = distance ÷ speed = \(\dfrac{1.5 \times 10^{8}}{3 \times 10^{5}} = 0.5 \times 10^{3} = 500\) seconds. ✓
\(500 \div 60 = 8.33\) minutes (3 s.f.). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.