Cambridge IGCSE Mathematics 0580: Powers, roots and indices I – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover 1.3 Powers and roots and 1.7 Indices I from the Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027. Section 1.3 and integer indices are Core and Extended. Fractional indices are Extended only, so those questions are labelled (Extended). Questions marked (non-calculator) match Paper 1 or Paper 2 conditions; questions marked (calculator) match Paper 3 or Paper 4.

Learn the methods first in the powers, roots and indices study guide. The 0580 course hub and the printable 0580 checklist show where this unit sits. The Number practice questions contain further mixed questions on Topic 1; the questions below are different.

Questions

1. (non-calculator) Write down the value of

(a) 15² [1] (b) ∛27 [1] (c) 4³ [1]

2. (non-calculator) Work out

(a) √(10² − 6²) [2] (b) 3 × ∛125 − 2³ [2]

3. (non-calculator) Work out

(a) (−2)⁵ [1] (b) 0.4² [1] (c) √0.49 [1] (d) (2/3)³ [1]

4. (calculator)

(a) Work out 1.35⁴. Write down all the figures on your calculator display. [1] (b) Work out the fifth root of 0.0761. Give your answer correct to 3 significant figures. [2] (c) Work out ∛(6.2² + 4.8). Give your answer correct to 3 significant figures. [2]

5. (non-calculator) Find the value of

(a) 6⁰ [1] (b) 2⁻⁵ [1] (c) (1/3)⁻² [1] (d) 10⁻², written as a decimal [1]

6. (non-calculator) Simplify each expression, leaving your answer as a power of 5.

(a) 5⁷ × 5⁻³ [1] (b) 5⁸ ÷ 5¹¹ [1] (c) (5⁻²)⁴ [1] (d) Find the value of (5⁶ × 5⁻⁴) ÷ 5⁵. Give your answer as a fraction. [2]

7. (non-calculator)

(a) Write 1/16 as a power of 2. [1] (b) Write 8² × 4⁻³ ÷ 16 as a single power of 2. [3]

8. (Extended) (non-calculator) Find the value of

(a) 49^(1/2) [1] (b) 1000^(2/3) [2] (c) 16^(−3/4) [2] (d) (9/25)^(−1/2) [2]

9. (Extended) (non-calculator)

(a) Find the value of 9^(3/2) × 27^(−2/3). [3] (b) Write √2 × 8 as a single power of 2. [2] (c) Show that 4^(−1/2) × 32^(2/5) = 2. [2]

10. (calculator)

(a) A square has an area of 70.56 cm². Calculate its perimeter. [2] (b) A cube has a volume of 42.875 cm³. Calculate the area of one face of the cube. [2] (c) Work out √(4.6² + 3.9²) ÷ 1.7³. Give your answer correct to 3 significant figures. [2] (d) Write these numbers in order of size, starting with the smallest.

2⁻³ 0.4² √0.0225 3⁻² [2]

11. (Extended) (non-calculator)

(a) Find the value of 125^(−1/3) + 8⁰. [2] (b) Write (√8 × 4⁻¹) ÷ 2^(1/2) as a single power of 2. [3] (c) Ravi says that 36^(−1/2) = −6. Explain his error and give the correct value. [2] (d) Find the value of 0.25^(−3/2). [2]

Answers

1. (a) 225 [1] (b) 3 [1] (c) 64 [1] Examiner insight: “Write down” means no working is expected, so the mark depends entirely on accurate recall; 4³ = 12 or ∛27 = 9 scores zero.

2. (a) 10² − 6² = 100 − 36 = 64 [1]. √64 = 8 [1]. (b) ∛125 = 5 and 2³ = 8 [1]. 3 × 5 − 8 = 7 [1]. Examiner insight: The method mark in (a) is for working out the inside of the root first; writing √10² − √6² = 10 − 6 = 4 earns nothing because the method is wrong.

3. (a) −32 [1] (b) 0.16 [1] (c) 0.7 [1] (d) 8/27 [1] Examiner insight: Each part is a single accuracy mark, so 32 (sign lost), 1.6 (decimal places miscounted) or 0.07 all score zero with no partial credit.

4. (a) 3.32150625 [1] (b) 0.0761^(1/5) = 0.597416… [1] = 0.597 (3 s.f.) [1]. (c) 6.2² + 4.8 = 43.24 [1]. ∛43.24 = 3.50990… = 3.51 (3 s.f.) [1]. Examiner insight: Writing the unrounded value before rounding protects you: a correct long display usually earns the method mark even if the final rounding is wrong, while a bare wrong 3 s.f. answer earns nothing.

5. (a) 1 [1] (b) 1/2⁵ = 1/32 [1] (c) (3/1)² = 9 [1] (d) 1/100 = 0.01 [1] Examiner insight: A negative index means reciprocal, not negative: answers of −32 in (b) or −9 in (c) score zero, and 0 or 6 in (a) also score zero.

6. (a) 5^(7 + (−3)) = 5⁴ [1] (b) 5^(8 − 11) = 5⁻³ [1] (c) 5^(−2 × 4) = 5⁻⁸ [1] (d) 5⁶ × 5⁻⁴ ÷ 5⁵ = 5^(6 − 4 − 5) = 5⁻³ [1] = 1/125 [1]. Examiner insight: When the question says “as a power of 5”, evaluating instead (for example 625 in (a)) does not answer the question as set; in (d) the instruction asks for a fraction, so 5⁻³ alone gets only the method mark.

7. (a) 2⁻⁴ [1] (b) 8² = (2³)² = 2⁶ [1]. 4⁻³ = (2²)⁻³ = 2⁻⁶ and 16 = 2⁴ [1]. 2⁶ × 2⁻⁶ ÷ 2⁴ = 2^(6 − 6 − 4) = 2⁻⁴ [1]. Examiner insight: Method marks come from rewriting each number with base 2; combining 8² × 4⁻³ as 32⁻¹ or similar by mixing bases gains no credit.

8. (a) √49 = 7 [1] (b) ∛1000 = 10 [1]. 10² = 100 [1]. (c) Fourth root of 16 = 2, so 16^(3/4) = 2³ = 8 [1]. Negative index gives 1/8 [1]. (d) (9/25)^(1/2) = 3/5 [1]. Reciprocal: 5/3 [1]. Examiner insight: The first mark in (b) to (d) is for a correct root; an answer of 8 in (c) shows the root and power but misses the reciprocal, so it scores one of the two marks.

9. (a) 9^(3/2) = (√9)³ = 27 [1]. 27^(−2/3) = 1/(∛27)² = 1/9 [1]. 27 × 1/9 = 3 [1]. (b) √2 = 2^(1/2) and 8 = 2³ [1]. 2^(1/2 + 3) = 2^(7/2) [1]. (c) 4^(−1/2) = 1/√4 = 1/2 [1]. 32^(2/5) = (fifth root of 32)² = 2² = 4, and 1/2 × 4 = 2 [1]. Examiner insight: In a “show that” the answer is given, so every value must be written down; writing 1/2 × 4 = 2 without showing where 4 came from loses the second mark.

10. (a) √70.56 = 8.4 cm [1]. Perimeter = 4 × 8.4 = 33.6 cm [1]. (b) ∛42.875 = 3.5 cm [1]. Area of one face = 3.5² = 12.25 cm² [1]. (c) √(21.16 + 15.21) = √36.37 = 6.03075… and 1.7³ = 4.913 [1]. 6.03075… ÷ 4.913 = 1.22750… = 1.23 (3 s.f.) [1]. (d) Values: 2⁻³ = 0.125, 0.4² = 0.16, √0.0225 = 0.15, 3⁻² = 0.111… [1]. Order: 3⁻², 2⁻³, √0.0225, 0.4² [1]. Examiner insight: In (d) the final mark needs the original forms in the correct order; converting each value to a decimal first earns the method mark even if one value is wrong.

11. (a) 125^(−1/3) = 1/∛125 = 1/5 and 8⁰ = 1 [1]. 1/5 + 1 = 6/5 (or 1 1/5 or 1.2) [1]. (b) √8 = 2^(3/2) [1]. 4⁻¹ = 2⁻² [1]. 2^(3/2 − 2 − 1/2) = 2⁻¹ [1]. (c) A negative index means the reciprocal, not a negative number [1]. 36^(−1/2) = 1/√36 = 1/6 [1]. (d) 0.25 = 1/4, so 0.25^(−3/2) = 4^(3/2) [1] = (√4)³ = 8 [1]. Examiner insight: An explanation mark in (c) needs a reason, not just the right value; “he is wrong, it is 1/6” scores only one of the two marks because it does not say what the negative index means.

Where marks are usually lost

• Writing a negative index as a negative answer: 2⁻⁵ is 1/32, not −32.
• Giving 0 or the base itself for a zero index: 6⁰ = 1.
• Multiplying indices when they should be added: 5⁷ × 5⁻³ = 5⁴, not 5⁻²¹.
• Using index rules with different bases without converting first, such as 8² × 4⁻³ treated as 32 to some power.
• Losing a sign when subtracting a negative index: 6⁻² ÷ 6⁻⁴ = 6², not 6⁻⁶.
• Square-rooting each term separately inside √(a² + b²).
• Miscounting decimal places in powers of decimals: 0.4² = 0.16 and √0.49 = 0.7.
• Rounding to 3 s.f. too early in a calculator question, or dropping a final zero (3.20 written as 3.2).
• (Extended) Doing the power before the root with large bases, or forgetting the reciprocal for a negative fractional index.

Next steps

• Read the powers, roots and indices study guide for worked explanations of every method used above.
• Revise the wider topic with the Number revision notes and Number study guide.
• Try more Topic 1 questions in the Number (Extended) practice questions.
• Take the 0580 Core diagnostic or the 0580 Extended diagnostic.
• Visit the 0580 course hub and tick off the printable 0580 checklist.
• Book a free trial class.

Official syllabus

Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027, Cambridge International Education. Subject content sections C1.3 and E1.3 (Powers and roots) and C1.7 and E1.7 (Indices I).