Chapter focus
A square number counts objects arranged in a square array; a cube number counts unit cubes arranged into a larger cube. This chapter uses patterns and factors to explore square roots and cube roots.
These original notes support revision and practice. Follow the diagrams, activities and full exercise sequence in your assigned NCERT edition as well.
1. Square numbers and their patterns
The square of a whole number n is n × n, written n2. It is the number of small square tiles in an n-by-n square array. The first few are 1, 4, 9, 16, 25, 36, … . A number such as 25 is a perfect square because 25 = 52.
Look at the change between consecutive squares: 1, then +3 gives 4, then +5 gives 9, then +7 gives 16. In general, (n + 1)2 − n2 = 2n + 1, so each new square is made by adding the next odd number. This pattern is a useful check, but it does not mean every odd number is itself a square.
For example, 1 + 3 + 5 + 7 = 16 = 42. Adding the first four odd numbers builds a square of side 4.
2. Cubes and cube numbers
The cube of a whole number n is n × n × n, written n3. It represents the number of unit cubes in a cube with n unit cubes along each edge. The first cubes are 1, 8, 27, 64, 125, … because 23 = 8, 33 = 27, and 53 = 125.
Square and cube are different operations: 42 = 16, while 43 = 64. Their inverse operations are the square root and cube root. Since 72 = 49, √49 = 7; since 43 = 64, ∛64 = 4.
Square roots of positive numbers have a positive principal value in this school context. Thus √49 means 7, although both 7 × 7 and (−7) × (−7) equal 49. A cube root keeps the sign: ∛(−8) = −2 because (−2)3 = −8.
3. Use prime factors to recognise perfect powers
Write a number as a product of primes. For a perfect square, every prime factor must occur an even number of times, so the exponents in its prime factorisation are even. For a perfect cube, each exponent must be a multiple of 3.
Example: Is 900 a perfect square?
- Factor: 900 = 9 × 100 = 32 × 22 × 52.
- Each exponent is even, so pair equal prime factors: (2 × 3 × 5)2.
- Therefore 900 = 302 and √900 = 30.
Example: Is 216 a perfect cube? Since 216 = 23 × 33 = (2 × 3)3 = 63, it is a perfect cube and ∛216 = 6.
This method also finds the least multiplier needed to make a perfect power. In 72 = 23 × 32, the exponent of 2 is odd. Multiplying by 2 gives 24 × 32 = 144 = 122; therefore 2 is the least positive integer multiplier that makes 72 a square.
4. Estimate a root before calculating
Use nearby perfect powers to check whether a proposed root is reasonable. Because 102 = 100 and 112 = 121, √115 lies between 10 and 11. It is closer to 11 because 115 is 6 below 121 and 15 above 100. An estimate is not an exact value; state that distinction clearly.
For cube roots, compare nearby cubes. Since 43 = 64 and 53 = 125, ∛90 lies between 4 and 5. Check a root by cubing it, or by bracketing it between known cubes.
Common error: √(a + b) is generally not √a + √b. For instance, √(9 + 16) = √25 = 5, while √9 + √16 = 3 + 4 = 7.
Worked examples
A. Find √1764 by prime factorisation
- 1764 = 2 × 882 = 22 × 441.
- 441 = 21 × 21 = 32 × 72, so 1764 = 22 × 32 × 72.
- Take one factor from each equal pair: √1764 = 2 × 3 × 7 = 42.
- Check: 42 × 42 = 1764.
B. Find the least number to multiply 432 by to make a cube
- 432 = 16 × 27 = 24 × 33.
- For a cube, each exponent must be a multiple of 3. The exponent 4 needs 2 more; the exponent 3 is already suitable.
- Multiply by 22 = 4. Then 432 × 4 = 26 × 33 = 123 = 1728.
Quick revision
n2 = n × n; consecutive squares differ by successive odd numbers.
n3 = n × n × n; ∛ reverses cubing.
Square: all exponents even. Cube: all exponents divisible by 3.
Practice: explain your reasoning
1. Is 2025 a perfect square? Find its square root.
Answer: 2025 = 45 × 45 = 452, so √2025 = 45.
2. Find the least positive integer that must multiply 540 to make a perfect square.
Answer: 540 = 22 × 33 × 5. The exponents of 3 and 5 are odd. Multiply by 3 × 5 = 15: 540 × 15 = 22 × 34 × 52 = 902. The least multiplier is 15.
3. Find ∛3375 by prime factorisation.
Answer: 3375 = 27 × 125 = 33 × 53 = 153, so the cube root is 15.
4. A square courtyard has area 784 m². What is its side length, and how much fencing is needed around it?
Answer: Side = √784 = 28 m. Perimeter = 4 × 28 = 112 m, so 112 m of fencing is needed if the whole boundary is fenced.
5. A student says 1000 is a perfect square because it ends in three zeroes. Is the claim correct?
Answer: No. 1000 = 23 × 53; its prime exponents are not all even. It is a perfect cube (103), not a perfect square.
Chapter summary
Square and cube numbers describe equal-sided arrays and solids. Patterns help estimate roots; prime factorisation gives a reliable test and exact method. For a perfect square, pair every prime factor; for a perfect cube, group them in threes. Always check a calculated root by squaring or cubing it.