A Square and A Cube
“A Square and A Cube” brings squares and cubes together. Squaring a number multiplies it by itself (5² = 25) and cubing multiplies it three times (5³ = 125). This Class 8 Ganita Prakash chapter explores the patterns hidden in square and cube numbers and shows how to find square roots and cube roots, most reliably by prime factorisation.
Learning objectives
- Recognise perfect squares and perfect cubes.
- Use patterns and properties of square numbers.
- Find square roots by prime factorisation.
- Find cube roots of perfect cubes by prime factorisation.
Key concepts
Square numbers
A number multiplied by itself gives its square, so 6² = 36. Numbers like 1, 4, 9, 16, 25 are perfect squares. A perfect square always ends in 0, 1, 4, 5, 6 or 9 — never in 2, 3, 7 or 8 — and the number of zeros at the end of a perfect square is always even.
Patterns in squares
Square numbers hide neat patterns: the sum of the first n odd numbers is n² (1 + 3 + 5 = 9 = 3²), and between consecutive squares n² and (n+1)² there are exactly 2n non-square numbers. These patterns help estimate and check squares quickly.
Square roots
A square root undoes squaring: since 7² = 49, √49 = 7. For perfect squares, write the number as a product of primes, pair the equal factors, and take one factor from each pair. For 144 = 2×2×2×2×3×3, the pairs give √144 = 2×2×3 = 12.
Cubes and cube roots
Cubing multiplies a number three times, so 4³ = 64; numbers like 1, 8, 27, 64, 125 are perfect cubes. To find a cube root, factor into primes and group the equal factors in threes, taking one from each group. For 216 = 2×2×2×3×3×3, ∛216 = 2×3 = 6.
Important formulas
Square
square of n = n × n = n²
Cube
cube of n = n × n × n = n³
Sum of odd numbers
1 + 3 + 5 + … + (2n − 1) = n²
Square root (prime factors)
√(perfect square) = product of one factor from each equal pair
Cube root (prime factors)
∛(perfect cube) = product of one factor from each equal triple
Key definitions
- Perfect square
- A number that is the square of a whole number, e.g. 49 = 7².
- Square root
- A number which, when multiplied by itself, gives the original number.
- Perfect cube
- A number that is the cube of a whole number, e.g. 27 = 3³.
- Cube root
- A number which, when multiplied by itself three times, gives the original number.
Solved examples
Q1. Find √324 by prime factorisation.
Solution: 324 = 2×2×3×3×3×3. Pairs: (2×2),(3×3),(3×3). Take one from each pair: 2×3×3 = 18. So √324 = 18.
Q2. Is 1000 a perfect square? Why?
Solution: 1000 ends in three zeros (an odd number of zeros), so it cannot be a perfect square.
Q3. Find ∛512.
Solution: 512 = 2⁹ = (2×2×2)×(2×2×2)×(2×2×2). One factor from each triple: 2×2×2 = 8. So ∛512 = 8.
Q4. Express 25 as a sum of consecutive odd numbers.
Solution: Since 25 = 5², it is the sum of the first 5 odd numbers: 1 + 3 + 5 + 7 + 9 = 25.
Common mistakes to avoid
- Assuming any number ending in 5 or 6 is a perfect square — the ending is necessary but not sufficient.
- Pairing factors in twos when finding a cube root (cube roots group in threes).
- Confusing squaring with doubling (5² = 25, not 10).
- Forgetting that a perfect square ends with an even number of zeros.
A Square and A Cube — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
The square of 9 is:
Practice questions
Short answer
How can you tell a number is not a perfect square from its last digit?
If it ends in 2, 3, 7 or 8 it cannot be a perfect square.
State the odd-number pattern for squares.
The sum of the first n odd numbers equals n².
How do you find a cube root by prime factorisation?
Factor into primes, group equal factors in threes, and take one factor from each group.
Long answer
Find the square root of 1764 by prime factorisation, showing each step.
1764 = 2 × 2 × 3 × 3 × 7 × 7. Make pairs: (2×2), (3×3), (7×7). Take one number from each pair: 2 × 3 × 7 = 42. Therefore √1764 = 42.
Explain why 243 is not a perfect cube, then find the smallest number to multiply it by to make one.
243 = 3⁵ = (3×3×3) × 3 × 3. One complete triple gives a 3, but two 3s are left over, so 243 is not a perfect cube. Multiplying by 3 makes 3⁶ = 729 = 9³, a perfect cube; so the smallest multiplier is 3.
HOTS (Higher Order Thinking)
Without computing, explain why 768 cannot be a perfect square.
768 = 2⁸ × 3 = (pairs of 2) × 3. The single 3 has no pair, so the factorisation cannot be split fully into equal pairs; hence 768 is not a perfect square.
A square garden has area 529 m². Find its side and explain.
The side is √529. Since 23² = 529, the side is 23 m — the side of a square equals the square root of its area.
Quick revision
Revision notes
- Square: n²; perfect squares end in 0,1,4,5,6,9 with an even number of trailing zeros.
- Sum of first n odd numbers = n².
- Square root: pair equal prime factors, take one per pair.
- Cube: n³; cube root: group equal prime factors in threes.
Key takeaways
- Prime factorisation is the reliable way to find square and cube roots.
- Digit and zero patterns quickly rule out non-squares.
- Squares pair factors in twos; cubes group them in threes.
Frequently asked questions
What is a perfect square?
A number that equals the square of a whole number, such as 36 = 6².
How is a cube root different from a square root?
A square root undoes squaring (pairs of factors); a cube root undoes cubing (groups of three equal factors).
Can a perfect square end in 8?
No — perfect squares only end in 0, 1, 4, 5, 6 or 9.