IGCSE Maths 0580 Powers and Roots Past Paper Questions and Worked Solutions

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IGCSE Maths (0580) · Number · 1.3

Every powers and roots question from 11 Cambridge IGCSE Mathematics (0580) past papers (May/June 2025 to May/June 2026), syllabus section 1.3. 12 Core-tier questions. Each question is labelled with its question paper code (for example 0580/12/M/J/26) and every final answer matches the official Cambridge mark scheme. Try each question first, then click “Show solution” to check your working. This page is part of our IGCSE Maths 0580 Number past paper questions.

0580 Powers and Roots past paper questions: Core (Papers 1 and 3)

11 markCore

May/June 2026 · Paper 1 · 0580/12/M/J/26 · Q1

Write down the value of √81.

Show solution
  1. √81 asks “which number times itself makes 81?”
  2. 9 × 9 = 81, so √81 = 9
Answer 9
21 markCore

May/June 2026 · Paper 1 · 0580/13/M/J/26 · Q18(a)

Write down the value of ∛27.

Show solution
  1. ∛27 asks “which number, multiplied by itself three times, makes 27?”
  2. 3 × 3 × 3 = 27, so ∛27 = 3
Answer 3
31 markCore

May/June 2026 · Paper 3 · 0580/31/M/J/26 · Q2(b)

Calculate 17.4².

Show solution
  1. Squaring means multiplying the number by itself: 17.4 × 17.4.
  2. Use a calculator: 17.4² = 302.76
Answer 302.76
44 marksCore

Feb/Mar 2026 · Paper 1 · 0580/12/F/M/26 · Q9

(a) Work out the difference between 5² and 2³. (b) Write down the reciprocal of 4. (c) Write down an irrational number that is between 4 and 5.

Show solution
Part (a)
  1. 5² = 5 × 5 = 25 and 2³ = 2 × 2 × 2 = 8.
  2. Difference = 25 − 8 = 17
Answer 17
Part (b)
  1. The reciprocal of a number is 1 divided by it.
  2. Reciprocal of 4 = 1⁄4 (or 0.25)
Answer 1⁄4
Part (c)
  1. 4 = √16 and 5 = √25, so any square root between √16 and √25 lies between 4 and 5.
  2. The square root of a number that is not a perfect square is irrational, for example √20
Answer √20
51 markCore

Feb/Mar 2026 · Paper 1 · 0580/12/F/M/26 · Q13

Work out √0.0025.

Show solution
  1. 0.0025 = 25⁄10000.
  2. √25 = 5 and √10000 = 100, so √0.0025 = 5⁄100 = 0.05
  3. Check: 0.05 × 0.05 = 0.0025 ✓
Answer 0.05
62 marksCore

Oct/Nov 2025 · Paper 1 · 0580/11/O/N/25 · Q17

(a) Write down a prime number between 20 and 30. (b) Simplify (√9)².

Show solution
Part (a)
  1. Check the numbers from 21 to 29: 21 = 3 × 7, 22 is even, 25 = 5 × 5, 27 = 3 × 9. 23 and 29 have only two factors.
  2. A prime number: 23 (or 29)
Answer 23
Part (b)
  1. √9 = 3, and 3² = 9. Squaring undoes the square root.
  2. (√9)² = 9
Answer 9
72 marksCore

Oct/Nov 2025 · Paper 1 · 0580/12/O/N/25 · Q4

(a) Write down the value of √144. (b) Work out the value of 5³.

Show solution
Part (a)
  1. √144 asks “which number times itself makes 144?” 12 × 12 = 144.
  2. √144 = 12
Answer 12
Part (b)
  1. 5³ = 5 × 5 × 5 = 25 × 5 = 125
Answer 125
82 marksCore

Oct/Nov 2025 · Paper 1 · 0580/13/O/N/25 · Q6

Anna thinks of a number, she multiplies it by 2 and then finds the square root. The answer is 12. What number did Anna think of?

Show solution
  1. Work backwards and undo each step in reverse order.
  2. The last step was a square root, so undo it by squaring: 12² = 144.
  3. Before that she multiplied by 2, so undo it by dividing: 144 ÷ 2 = 72.
  4. Anna’s number = 72
  5. Check: 72 × 2 = 144 and √144 = 12 ✓
Answer 72
91 markCore

Oct/Nov 2025 · Paper 3 · 0580/32/O/N/25 · Q7

Calculate √361 + 1.

Show solution
  1. Find the square root first: 19 × 19 = 361, so √361 = 19.
  2. Then add 1: 19 + 1 = 20
Answer 20
102 marksCore

Oct/Nov 2025 · Paper 3 · 0580/33/O/N/25 · Q6

(a) Write down the reciprocal of 8. (b) Work out 19³.

Show solution
Part (a)
  1. The reciprocal of a number is 1 divided by that number.
  2. Reciprocal of 8 = 1⁄8 (= 0.125)
Answer 1⁄8
Part (b)
  1. 19³ = 19 × 19 × 19.
  2. 19 × 19 = 361, and 361 × 19 = 6859
Answer 6859
112 marksCore

May/June 2025 · Paper 1 · 0580/12/M/J/25 · Q3

Write down the value of (a) √36 (b) 10³.

Show solution
Part (a)
  1. √36 asks “which number times itself makes 36?” 6 × 6 = 36.
  2. √36 = 6
Answer 6
Part (b)
  1. 10³ = 10 × 10 × 10 = 1000
Answer 1000
125 marksCore

May/June 2025 · Paper 1 · 0580/13/M/J/25 · Q3

(a) Write down √169.

(b) Work out the value of 2⁴.

(c) Work out the value of 10−2.

(d) Work out −18 ÷ −4.

(e) Work out 1.6 × 0.02.

Show solution
Part (a)
  1. √169 asks “which number times itself makes 169?” 13 × 13 = 169.
  2. √169 = 13
Answer 13
Part (b)
  1. 2⁴ = 2 × 2 × 2 × 2 = 4 × 4 = 16
Answer 16
Part (c)
  1. A negative power means “1 over”: 10−2 = 1⁄10² = 1⁄100.
  2. 1⁄100 = 0.01
Answer 1⁄100 = 0.01
Part (d)
  1. Negative ÷ negative = positive, so work out 18 ÷ 4.
  2. 18 ÷ 4 = 4.5, so −18 ÷ −4 = 4.5
Answer 4.5
Part (e)
  1. Ignore the decimal points first: 16 × 2 = 32.
  2. Count the decimal places in the question: 1.6 has 1 and 0.02 has 2, so the answer has 3.
  3. 1.6 × 0.02 = 0.032
Answer 0.032

Question papers used on this page

All powers and roots questions above, listed by 0580 question paper code.

QP codeFile namePaperQuestions
0580/12/M/J/260580_s26_qp_12May/June 2026 · Paper 1 (Core)Q1
0580/13/M/J/260580_s26_qp_13May/June 2026 · Paper 1 (Core)Q18(a)
0580/31/M/J/260580_s26_qp_31May/June 2026 · Paper 3 (Core)Q2(b)
0580/12/F/M/260580_m26_qp_12Feb/Mar 2026 · Paper 1 (Core)Q9, Q13
0580/11/O/N/250580_w25_qp_11Oct/Nov 2025 · Paper 1 (Core)Q17
0580/12/O/N/250580_w25_qp_12Oct/Nov 2025 · Paper 1 (Core)Q4
0580/13/O/N/250580_w25_qp_13Oct/Nov 2025 · Paper 1 (Core)Q6
0580/32/O/N/250580_w25_qp_32Oct/Nov 2025 · Paper 3 (Core)Q7
0580/33/O/N/250580_w25_qp_33Oct/Nov 2025 · Paper 3 (Core)Q6
0580/12/M/J/250580_s25_qp_12May/June 2025 · Paper 1 (Core)Q3
0580/13/M/J/250580_s25_qp_13May/June 2025 · Paper 1 (Core)Q3

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