IGCSE Mathematics: Squares, Cubes and Roots Practice Questions
A square number is the result of multiplying an integer by itself, and a cube number of multiplying it three times. Roots reverse those operations, and any equation containing a squared term has two solutions.
Topic 1.3 of Cambridge IGCSE Mathematics 0580 covers powers and roots as numerical values, and appears mostly in the non-calculator sections. The habit worth building is remembering that a squared equation has two answers. Index laws applied to algebraic terms are a separate topic, covered on the index laws page.
What you need to know for Powers and Roots
- PowersThe index tells you how many times the base is multiplied by itself. 25 means 2 x 2 x 2 x 2 x 2 = 32.
- RootsThe square root of 144 is 12 because 12 x 12 = 144. The cube root of 64 is 4 because 4 x 4 x 4 = 64.
- Two square rootsThe square root symbol means the positive root only, but the equation x2 = 81 has two solutions, x = 9 and x = minus 9.
- Order of operationsPowers and roots are evaluated before multiplication and division, and well before addition and subtraction. Brackets override that order.
- Recognising square and cube numbersSquares to 15: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Cubes to 5: 1, 8, 27, 64, 125.
- Estimating a rootLocate the number between two known squares or cubes. The square root of 70 lies between 8 and 9, because 82 = 64 and 92 = 81.
IGCSE Mathematics Powers and Roots questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Work out, without a calculator, the value of 53, the square root of 144 and the cube root of 64.
Show the worked answer
- 53 means 5 x 5 x 5. 5 x 5 = 25, and 25 x 5 = 125.
- The square root of 144 is the number that multiplies by itself to give 144. Since 12 x 12 = 144, the square root is 12.
- The cube root of 64 is the number that cubes to give 64. Since 4 x 4 x 4 = 64, the cube root is 4.
- Learning squares to 15 and cubes to 5 makes these instant.
Solve the equation x2 = 81, giving both solutions. Then explain why the square root of 81 is written as 9 only.
Show the worked answer
- Take the square root of both sides. Both 9 x 9 and minus 9 multiplied by minus 9 give 81.
- So x = 9 or x = minus 9. Both must be stated.
- The square root symbol is defined to give the principal, positive root only, so the square root of 81 is written as 9.
- When solving an equation you must include both, but when evaluating the root symbol you take only the positive value.
Work out the value of 62 minus 43 plus the square root of 81, without a calculator.
Show the worked answer
- Evaluate each power and root before adding or subtracting.
- 62 = 36 and 43 = 64.
- The square root of 81 is 9.
- 36 minus 64 plus 9 = minus 28 plus 9 = minus 19.
Without using a calculator, state which two consecutive whole numbers the square root of 70 lies between, and explain your reasoning.
Show the worked answer
- Find the square numbers on either side of 70.
- 82 = 64, which is less than 70.
- 92 = 81, which is greater than 70.
- Since 64 is less than 70 which is less than 81, the square root of 70 lies between 8 and 9. It is closer to 8, since 70 is nearer to 64 than to 81.
Common mistakes in this topic
- Treating an index as a multiplier, so 43 becomes 12.
- Working left to right instead of evaluating powers and roots first.
- Giving only the positive solution when solving a squared equation.
- Applying index laws to terms with different bases.
- Forgetting that any non-zero number to the power 0 equals 1.
Exam tips
- Memorise squares to 15 and cubes to 5. It saves time across the whole paper, not just in this topic.
- Write the index law you are using next to your working. The rule itself often carries a method mark.
- When an equation contains a squared term, expect two solutions and check both make sense in context.
- For estimating roots, always name the two square or cube numbers you are working between.
- A negative index means a reciprocal, not a negative answer.
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Powers and Roots FAQs
What are the square numbers I need to know for IGCSE?
Learn the squares of 1 to 15, which are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196 and 225, together with the cubes of 1 to 5, which are 1, 8, 27, 64 and 125. Recognising them instantly saves time across the whole paper.
Does a square root have one answer or two?
The square root symbol denotes the positive root only, so the square root of 81 is written as 9. However, the equation x squared equals 81 has two solutions, 9 and minus 9, because both give 81 when squared. Always give both when solving an equation.
How do I estimate a square root without a calculator?
Find the two square numbers the value lies between. Since 64 is less than 70 and 70 is less than 81, the square root of 70 lies between 8 and 9. Compare how close the number is to each square to judge which whole number it is nearer.
What does a negative index mean?
A negative index means take the reciprocal of the positive power. So 2 to the power minus 3 equals 1 divided by 2 cubed, which is one eighth. The answer is a fraction, not a negative number, which is a frequent source of confusion.
This page covers powers and roots as numbers. For the index laws applied to algebraic terms, including negative and fractional indices, see Indices.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.