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 |

**Essential values to memorise:**

| n | n² | n³ | |---|----|----| | 1 | 1 | 1 | | 2 | 4 | 8 | | 3 | 9 | 27 | | 4 | 16 | 64 | | 5 | 25 | 125 | | 6 | 36 | 216 | | 7 | 49 | 343 | | 8 | 64 | 512 | | 9 | 81 | 729 | | 10 | 100 | 1000 | | 11 | 121 | 1331 | | 12 | 144 | 1728 | | 13 | 169 | – | | 14 | 196 | – | | 15 | 225 | – |

Worked Examples

**Example 1: Find the square root of 1764 using prime factorisation.**

Step 1: Factorise 1764

• 1764 ÷ 2 = 882
• 882 ÷ 2 = 441
• 441 ÷ 3 = 147
• 147 ÷ 3 = 49
• 49 ÷ 7 = 7
• 7 ÷ 7 = 1

So, 1764 = 2 × 2 × 3 × 3 × 7 × 7 = 2² × 3² × 7²

Step 2: Take one factor from each pair √1764 = 2 × 3 × 7 = 42

**Answer: √1764 = 42**

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**Example 2: Find the cube root of 5832.**

Step 1: Factorise 5832

• 5832 ÷ 2 = 2916
• 2916 ÷ 2 = 1458
• 1458 ÷ 2 = 729
• 729 ÷ 3 = 243
• 243 ÷ 3 = 81
• 81 ÷ 3 = 27
• 27 ÷ 3 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

So, 5832 = 2³ × 3⁶ = 2³ × (3²)³

Step 2: Take one factor from each triplet ∛5832 = 2 × 3² = 2 × 9 = 18

**Answer: ∛5832 = 18**

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**Example 3: What is the smallest number by which 1152 must be multiplied to make it a perfect cube?**

Step 1: Factorise 1152 1152 = 2⁷ × 3²

Step 2: For a perfect cube, all exponents must be multiples of 3

• 2⁷ needs 2 more 2s to become 2⁹
• 3² needs 1 more 3 to become 3³

Step 3: Multiply by the missing factors Required number = 2² × 3 = 4 × 3 = 12

**Answer: Multiply by 12**

Common Mistakes

• **Confusing square and cube roots**: Students sometimes apply pairing method for cube roots or triplet method for square roots. Fix: Always remember—square root uses pairs (groups of 2), cube root uses triplets (groups of 3).

• **Forgetting that negative numbers have no real square root**: √(−25) is not defined in real numbers. Fix: Real square roots exist only for non-negative numbers.

• **Assuming all even numbers are perfect squares**: 18 is even but not a perfect square. Fix: Check if all prime factors appear an even number of times.

• **Incomplete factorisation**: Stopping before reaching all prime factors leads to wrong answers. Fix: Continue dividing until you reach 1.

• **Unit digit shortcuts applied incorrectly**: A number ending in 4 could be 4, 14, 24... but only some are perfect squares. Fix: Unit digit check eliminates possibilities (2, 3, 7, 8 endings are never perfect squares) but does not confirm a number is a perfect square.

Quick Reference

• Square of n = n × n; Cube of n = n × n × n

• Perfect square: every prime factor appears an even number of times

• Perfect cube: every prime factor appears in multiples of three

• Square root by factorisation: pair the primes, pick one from each pair

• Cube root by factorisation: group primes in threes, pick one from each group

• Numbers ending in 2, 3, 7 or 8 can never be perfect squares