Square Roots, Cube Roots and Factorisation

Overview

This topic forms a foundational pillar of arithmetic and number theory in the MAHA TET Mathematics section. Questions frequently test your ability to identify perfect squares and cubes, compute roots manually using factorisation, and apply these concepts in word problems involving area, volume and simplification.

For Paper I (Classes I–V), expect basic identification of squares and cubes up to 15, simple square root calculations, and recognition of perfect squares. Paper II (Classes VI–VIII) extends to prime factorisation method for finding roots, properties of squares and cubes, and their application in algebraic contexts. Mastery here also supports related topics like LCM-HCF, rational numbers and mensuration.

Students must be able to work without a calculator—exam questions reward those who have memorised key values and can apply systematic factorisation methods quickly and accurately.

Key Concepts

• **Square of a number**: When a number is multiplied by itself, the result is its square. Example: 7 × 7 = 49, so 49 is the square of 7.

• **Cube of a number**: When a number is multiplied by itself three times, the result is its cube. Example: 4 × 4 × 4 = 64, so 64 is the cube of 4.

• **Perfect square**: A number that can be expressed as the square of a whole number. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

• **Perfect cube**: A number that can be expressed as the cube of a whole number. Examples: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

• **Square root (√)**: The inverse operation of squaring. If n² = m, then √m = n. Example: √144 = 12.

• **Cube root (∛)**: The inverse operation of cubing. If n³ = m, then ∛m = n. Example: ∛216 = 6.

• **Prime factorisation method**: Breaking a number into prime factors helps find roots. For square roots, pair the prime factors; for cube roots, group them in triplets.

• **Property of perfect squares**: In the prime factorisation of a perfect square, every prime factor appears an even number of times.

• **Property of perfect cubes**: In the prime factorisation of a perfect cube, every prime factor appears in multiples of three.

Formulas / Key Facts

| Concept | Formula / Fact | |---------|----------------| | Square of n | n² = n × n | | Cube of n | n³ = n × n × n | | Square root | √(n²) = n | | Cube root | ∛(n³) = n | | Square root by factorisation | Write number as product of prime pairs, take one from each pair | | Cube root by factorisation | Write number as product of prime triplets, take one from each triplet | | Unit digit pattern for squares | 0→0, 1→1, 2→4, 3→9, 4→6, 5→5, 6→6, 7→9, 8→4, 9→1 | | Numbers ending in 2, 3, 7, 8 | Cannot be perfect squares | | Sum of first n odd numbers | 1 + 3 + 5 + ... + (2n−1) = n² |