A Square and A Cube | Mathematics Grade 8 CBSE

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The six ideas of squares and cubes

Squaring and cubing are two simple moves, but they come with their own names and shortcuts. Tap each term to see what it means and how to work it out.

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The three big ideas

A square number is a whole number multiplied by itself: n² = n × n. So 7² = 49 — exactly the number of unit squares in a 7 × 7 grid, which is why we call it "squared".
Perfect squares are 1, 4, 9, 16, 25, 36… — each is the square of a whole number.
Last-digit clue: a perfect square only ever ends in 0, 1, 4, 5, 6 or 9 — never 2, 3, 7 or 8. And the number of zeros at its end is always even.
The square root undoes squaring: √49 = 7 because 7² = 49. For a perfect square, find its prime factors and pair them up.

Worked example. Find √1296 using prime factorisation.

1. Break into primes: 1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 = 2⁴ × 3⁴.

2. Pair the primes: (2 × 2)(2 × 2)(3 × 3)(3 × 3) — take one from each pair.

3. √1296 = 2² × 3² = 4 × 9 = 36. Check: 36 × 36 = 1296. ✓

A cube number is a whole number multiplied by itself three times: n³ = n × n × n. So 4³ = 64 — the number of 1 cm cubes that fill a 4 × 4 × 4 cm box.
Perfect cubes are 1, 8, 27, 64, 125, 216… In their prime factorisation, every prime appears a number of times that is a multiple of 3.
The cube root undoes cubing: ∛64 = 4. Find the prime factors and group them in threes.
Unlike squares, a cube keeps the sign: a negative number cubed stays negative, so (−2)³ = −8.

Worked example. Find ∛3375 using prime factorisation.

1. Break into primes: 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3³ × 5³.

2. Group in threes: (3 × 3 × 3) and (5 × 5 × 5) — take one factor from each group.

3. ∛3375 = 3 × 5 = 15. Check: 15 × 15 × 15 = 3375. ✓

Common mistake: writing 6² as 6 × 2 = 12. Squaring means 6 × 6 = 36, and cubing means 6 × 6 × 6 = 216 — never multiply by the small "2" or "3".

Odd numbers build squares: 1 = 1², 1 + 3 = 2², 1 + 3 + 5 = 3²… The sum of the first n odd numbers is exactly n².
Gaps between squares: between n² and (n+1)² there are exactly 2n whole numbers that are not squares — the gaps grow as numbers get bigger.
Odd numbers also build cubes: 1 = 1³, 3 + 5 = 2³, 7 + 9 + 11 = 3³ — take the next 1, 2, 3… odd numbers.
Sum of cubes: 1³ + 2³ + 3³ + … + n³ = (1 + 2 + 3 + … + n)². For example 1³ + 2³ + 3³ = 36 = (1 + 2 + 3)² = 6².

Worked example. Use a pattern to add 1 + 3 + 5 + 7 + 9 + 11 + 13.

1. These are the first 7 odd numbers.

2. The sum of the first n odd numbers is n², so here it is 7².

3. 7² = 49 — no long addition needed.

Where you'll meet it

Squares and cubes at work

Tiling and flooring

To cover a square floor with square tiles and no cutting, you need the side length — the square root of the area. A 196 m² hall has a 14 m side, so 14 rows of 14 tiles fit exactly.

Packing and storage

A cube-shaped carton of side 5 boxes holds 5³ = 125 boxes. Cubes tell warehouses and sweet shops how much a stack holds, and cube roots tell them the edge length they need.

Finding distances

Square roots appear whenever lengths come from areas — the diagonal of a field, the height a ladder reaches, the shortest path across a plot. The square gives the area; the root gives the length back.

Check yourself

Competency quiz

Modelled on the competency-based pattern — MCQ, assertion–reason and case studies, testing whether you can use the ideas, not just recall them.

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