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Class 8 A Square and A Cube: cheat sheet
CBSE Mathematics. Perfect squares and cubes, their patterns, finding square and cube roots by prime factorisation, and estimating roots.
Key ideas
- A perfect square ends in 0, 1, 4, 5, 6 or 9, and has an even number of zeros at the end. A number ending in 2, 3, 7 or 8 is never a square.
- The sum of the first n odd numbers is n²: 1 + 3 + 5 + 7 = 16 = 4².
- Square root by prime factorisation: pair up equal prime factors and take one from each pair.
- Cube root: group equal prime factors in threes and take one from each group.
- To make a perfect square (or cube), multiply by the primes needed to complete every pair (or triple).
- Estimate a root between two known squares: 40 lies between 36 and 49, so √40 is between 6 and 7.
- 1729 = 1³ + 12³ = 9³ + 10³, the smallest number that is a sum of two cubes in two ways (the Hardy–Ramanujan number).
Remember
| Consecutive squares | (n + 1)² − n² = 2n + 1 |
| Roots of products | √(a²b²) = ab; ∛(a³b³) = ab |
| Example | 72 = 2³ × 3²: × 2 gives a square (144 = 12²); × 3 gives a cube (216 = 6³) |
| Cube of a negative | (−3)³ = −27, so ∛(−27) = −3 |
Watch out
- Thinking every number ending in 1, 4, 5, 6 or 9 is a square (11 and 14 are not).
- Taking one factor from each triple when finding a square root (use pairs for squares).
- Saying a negative number has no cube root.
Worked example
Find the smallest number by which 72 must be multiplied to make a perfect cube.
Working: 72 = 2³ × 3²: make every power a multiple of 3 → × 3 = 216 = 6³
Answer: 3
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