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Cambridge IGCSE Maths (0580) · Topic 1: Number · 1.3

Cambridge IGCSE Maths 0580 Powers and Roots Questions

Powers and Roots questions with worked answers for Cambridge IGCSE Mathematics (0580). Each question says whether it is Core or Extended and whether it is for a calculator or non-calculator paper. Try each one before you open the answer.

Practise powers and roots →Powers and Roots notes

What the syllabus asks

Summarised in our own words: check the exact wording in the official 0580 syllabus.

Powers and Roots questions: Core

For everyone: Core students (Papers 1 and 3) and Extended students (Papers 2 and 4).

Question 1 Core non-calculator · easy

Work out
(a) \(13^2\) (b) \(\sqrt{196}\) (c) \(\sqrt[3]{64}\) (d) \(2^5\)

Show the worked answer
  1. (a) \(13 \times 13 = 169\)
  2. (b) \(14 \times 14 = 196\), so \(\sqrt{196} = 14\)
  3. (c) \(4 \times 4 \times 4 = 64\), so \(\sqrt[3]{64} = 4\)
  4. (d) \(2 \times 2 \times 2 \times 2 \times 2 = 32\)

Answer: (a) \(169\) (b) \(14\) (c) \(4\) (d) \(32\)

Question 2 Core calculator · easy

Use your calculator to work out \(\sqrt{7.29} + 1.5^3\).

Show the worked answer
  1. \(\sqrt{7.29} = 2.7\)
  2. \(1.5^3 = 3.375\)
  3. \(2.7 + 3.375 = 6.075\)

Answer: \(6.075\)

Question 3 Core calculator · easy · 2 marks

Use your calculator to work out

(a) \(2.5^4\) (b) \(\sqrt[3]{50}\)

Give each answer correct to 3 significant figures.

Show the answer and mark scheme
  • B1 (a) 39.1 (39.0625)
  • B1 (b) 3.68 (3.684…)

Answer: (a) \(39.1\) (b) \(3.68\)

Question 4 Core non-calculator · medium

Find the two square numbers that add up to \(74\).

Show the worked answer
  1. Subtract each square number below \(74\) from \(74\): \(74 - 1 = 73\), \(74 - 4 = 70\), \(74 - 9 = 65\), \(74 - 16 = 58\), \(74 - 25 = 49\).
  2. \(49 = 7^2\) is a square number, so \(25 + 49 = 74\).

Answer: \(25\) and \(49\)

Question 5 Core non-calculator · medium

Between which two consecutive whole numbers does \(\sqrt{90}\) lie?

Show the worked answer
  1. \(9^2 = 81\) and \(10^2 = 100\).
  2. \(81 \lt 90 \lt 100\), so \(9 \lt \sqrt{90} \lt 10\).

Answer: \(9\) and \(10\)

Question 6 Core non-calculator · medium

Work out
(a) \(0.3^2\)
(b) \(\sqrt[3]{0.008}\)

Show the worked answer
  1. (a) \(0.3 \times 0.3 = 0.09\)
  2. (b) \(0.2 \times 0.2 \times 0.2 = 0.008\), so \(\sqrt[3]{0.008} = 0.2\)

Answer: (a) \(0.09\) (b) \(0.2\)

Question 7 Core calculator · medium

A square has an area of \(50\) cm². A cube has a volume of \(50\) cm³.
How much longer is a side of the square than an edge of the cube? Give your answer correct to 2 decimal places.

Show the worked answer
  1. Side of the square: \(\sqrt{50} = 7.0710\ldots\) cm.
  2. Edge of the cube: \(\sqrt[3]{50} = 3.6840\ldots\) cm.
  3. Difference: \(7.0710\ldots - 3.6840\ldots = 3.3870\ldots\) cm.

Answer: \(3.39\) cm

Question 8 Core calculator · medium · 2 marks

Work out \(2^5 - 3^4\).

Show the answer and mark scheme
  • M1 32 and 81 seen
  • A1 −49

Answer: \(-49\)

Question 9 Core calculator · medium · 2 marks

A square has an area of \(90\text{ cm}^2\). Find the length of one side. Give your answer correct to 3 significant figures.

Show the answer and mark scheme
  • M1 √90
  • A1 9.49 (9.486…)

Answer: \(9.49\) cm

Question 10 Core non-calculator · hard

Find the two smallest whole numbers greater than \(1\) that are both square numbers and cube numbers.

Show the worked answer
  1. A number that is both a square and a cube is a sixth power: \(n^6 = (n^3)^2 = (n^2)^3\).
  2. \(2^6 = 64 = 8^2 = 4^3\)
  3. \(3^6 = 729 = 27^2 = 9^3\)

Answer: \(64\) and \(729\)

Examiner Insights: common mistakes in IGCSE Maths 0580, with fixes (our analysis of public sources) →

Every 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. Questions with a mark scheme come from our 0580 practice bank (M method, A accuracy, B independent marks).

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Questions students ask

Which Cambridge 0580 papers can test powers and roots?

Powers and Roots is on both tiers (syllabus 1.3), so it can come up on Core Papers 1 (non-calculator) and 3 (calculator), and Extended Papers 2 (non-calculator) and 4 (calculator). Extended candidates also need the Extended parts listed on this page. Half of each tier's marks are on a non-calculator paper, so practise the non-calculator questions without one.

Are these Cambridge past-paper questions?

No. Every question here was written for IGCSE Math Revision in the style of the 0580 papers, and each answer was checked independently before publishing. For real exam practice, use the Cambridge past papers linked from our 0580 past papers page.

Where can I practise more powers and roots questions?

In the 0580 practice for topic 1 (Number), where every question is marked and tagged Core or Extended. Practice is part of the IGCSE plan, with a free preview; the questions on this page are free.

More number questions