Schooling · Class 8 · Mathematics · A Square and A Cube · Chapter on a page
CBSE Class 8 · Mathematics · NCERT Ganita Prakash Part-I
A Square and A Cube: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- square number
- cube number
- square root
- cube root
- Square numbers are always positive.…
- Cube numbers keep the sign of the…
- Not every number is a perfect square…
- Formulas and facts
- n² = n × n
- n³ = n × n × n
- √(n²) = n
- ∛(n³) = n
- First ten perfect squares
- First ten perfect cubes
- Pattern
- Watch out for
- Confusing square with doubling
- Forgetting that squaring a negative…
- Thinking cube root of a negative…
- Missing pairs during prime…
- Writing √25 = ±5 in basic problems
What this chapter is about
This chapter introduces you to two important ideas in mathematics: square numbers and cube numbers. You already know how to multiply numbers. When you multiply a number by itself, you get its square. When you multiply a number by itself twice (three copies of the number multiplied together), you get its cube.
Key ideas
- A square number is the result of multiplying a whole number by itself. For example, 5 × 5 = 25, so 25 is the square of 5, written as 5² = 25.
- A cube number is the result of multiplying a whole number by itself twice. For example, 4 × 4 × 4 = 64, so 64 is the cube of 4, written as 4³ = 64.
- The square root of a number is the value that, when multiplied by itself, gives that number. The square root of 36 is 6, because 6 × 6 = 36.
- The cube root of a number is the value that, when multiplied by itself twice, gives that number. The cube root of 27 is 3, because 3 × 3 × 3 = 27.
- Square numbers are always positive. The squares of both positive and negative numbers give positive results: (−3)² = 9 and 3² = 9.
- Cube numbers keep the sign of the original number: (−2)³ = −8 while 2³ = 8.
- Not every number is a perfect square or perfect cube. For instance, 20 is not a perfect square, and 50 is not a perfect cube.
Formulas and facts to remember
- Square of n: n² = n × n (area of a square with side n units)
- Cube of n: n³ = n × n × n (volume of a cube with side n units)
- Square root of n²: √(n²) = n
- Cube root of n³: ∛(n³) = n
- First ten perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
- First ten perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
- Pattern: The difference between consecutive squares increases by 2 each time (4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7, and so on)
Common mistakes
- Confusing square with doubling → 5² means 5 × 5 = 25, not 5 × 2 = 10.
- Forgetting that squaring a negative number gives a positive result → (−4)² = 16, not −16.
- Thinking cube root of a negative number does not exist → (−3)³ = −27, so ∛(−27) = −3.
- Missing pairs during prime factorisation → always group factors in pairs for square root, in triplets for cube root.
- Writing √25 = ±5 in basic problems → the principal square root is the positive value, so √25 = 5.
Quick revision
- n² means n times n; n³ means n times n times n.
- Square root reverses squaring; cube root reverses cubing.
- Use prime factorisation: pair factors for square root, group in threes for cube root.
- Squares are always positive; cubes carry the sign of the original number.
- Memorise squares up to 15 and cubes up to 10 for quick calculations.