Study Notes: Square & Square Roots

Overview

Square and square root problems appear reliably in the UPSSSC PET Elementary Arithmetic section, typically accounting for 2–3 questions. This topic tests your ability to identify perfect squares, extract square roots through standard methods, and apply these concepts to practical problems. Mastery here saves precious exam time because many percentage, average, and geometry problems depend on quick mental calculation of squares and roots.

The exam primarily focuses on three skills: (1) recognizing perfect squares up to 30² (900), (2) calculating square roots by long division for non-perfect squares, and (3) using prime factorisation for perfect squares. Unlike higher mathematics, you won't face irrational root approximations or complex algebraic manipulation—the emphasis is on computational accuracy and speed. Students who memorise squares 1–30 and understand the two extraction methods can confidently tackle these questions within 60–90 seconds each.

Key Concepts

• **Perfect Square**: A number obtained by multiplying an integer by itself. Examples: 1, 4, 9, 16, 25, 36... Perfect squares always end in 0, 1, 4, 5, 6, or 9; never in 2, 3, 7, or 8.

• **Square Root**: The inverse operation of squaring. If a² = b, then √b = a. Every positive number has two roots (positive and negative), but exam questions typically ask for the principal (positive) root.

• **Prime Factorisation Method**: Express the number as a product of prime factors. If all primes appear in pairs, the number is a perfect square. Take one factor from each pair to find the root.

• **Long Division Method**: A systematic algorithm for extracting square roots digit-by-digit, working left-to-right from the decimal point. Works for both perfect and non-perfect squares, producing decimal approximations when needed.

• **Unit Digit Pattern**: The last digit of a square depends only on the last digit of the original number: 1→1, 2→4, 3→9, 4→6, 5→5, 6→6, 7→9, 8→4, 9→1, 0→0. Use this to eliminate wrong answer choices quickly.

• **Property of Products**: √(a × b) = √a × √b. This simplifies calculation: √72 = √(36 × 2) = 6√2.

• **Square of Numbers Ending in 5**: For any number ending in 5, say 'a5', its square equals a(a+1) followed by 25. Example: 25² = 2×3 followed by 25 = 625; 85² = 8×9 followed by 25 = 7225.

• **Non-Perfect Square Identification**: If prime factorisation yields any prime with an odd exponent, the number is not a perfect square. Example: 72 = 2³ × 3¹ (both odd exponents) → not perfect.

Formulas / Key Facts

1. **Squares 1–30**: Memorise 1²=1, 2²=4, 3²=9... up to 30²=900. Also useful: 40²=1600, 50²=2500, 100²=10000.

2. **Sum of First n Odd Numbers**: 1 + 3 + 5 + ... + (2n-1) = n². Example: 1+3+5+7+9 = 25 = 5².

3. **(a + b)² = a² + 2ab + b²** and **(a - b)² = a² - 2ab + b²**. Useful for mental calculation: 98² = (100-2)² = 10000 - 400 + 4 = 9604.

4. **Between Consecutive Squares**: Between n² and (n+1)² there are exactly 2n non-square integers.

5. **Decimal Shifts**: √(100a) = 10√a; √(a/100) = (√a)/10. Moving the decimal two places multiplies/divides the root by 10.

6. **Prime Factorisation Result**: If n = p₁^(a₁) × p₂^(a₂) × ... is a perfect square, all exponents a₁, a₂, ... must be even. Then √n = p₁^(a₁/2) × p₂^(a₂/2) × ...

7. **Pythagorean Triples**: Common pairs like 3-4-5, 5-12-13, 8-15-17 help identify perfect squares quickly in geometry problems.

Worked Examples

**Example 1: Is 1764 a perfect square? If yes, find its square root by prime factorisation.**

Solution:

• Step 1: Factorise 1764.
• 1764 ÷ 2 = 882
• 882 ÷ 2 = 441
• 441 ÷ 3 = 147
• 147 ÷ 3 = 49
• 49 ÷ 7 = 7
• 7 ÷ 7 = 1
• Step 2: Write as powers: 1764 = 2² × 3² × 7²
• Step 3: All exponents are even, so 1764 is a perfect square.
• Step 4: √1764 = 2 × 3 × 7 = 42

**Example 2: Find √7056 using the long division method.**

Solution:

• Step 1: Group digits in pairs from right: 70|56
• Step 2: Largest square ≤70 is 64 (8²). Write 8, subtract: 70-64=6
• Step 3: Bring down 56 → 656
• Step 4: Double the quotient: 2×8=16. Find x such that 16x × x ≤ 656. Try x=4: 164×4=656 ✓
• Step 5: Quotient = 84
• Answer: √7056 = 84

**Example 3: Between which two consecutive integers does √200 lie?**

Solution:

• Step 1: Find nearest perfect squares.
• 14² = 196
• 15² = 225
• Step 2: Since 196 < 200 < 225, we have 14 < √200 < 15
• Answer: √200 lies between 14 and 15 (approximately 14.14)

Common Mistakes

1. **Forgetting to pair digits from the right in long division** → Incorrect grouping leads to wrong quotients. Always start pairing from the decimal point, moving outward. For 5329, group as 53|29, not 532|9.

2. **Assuming non-ending-in-square-digit means non-perfect-square** → While 2, 3, 7, 8 endings guarantee non-perfect, a 0, 1, 4, 5, 6, 9 ending doesn't guarantee perfection. Example: 52 ends in 2 (non-perfect ✓), but 51 ends in 1 yet isn't perfect. Always verify with factorisation or long division.

3. **Missing pairs in prime factorisation** → Students sometimes write √(2² × 3²) = 2 × 3 × √1 = 6√1 = 6 but forget to extract both factors. Correct approach: take one factor from each complete pair only.

4. **Doubling incorrectly in long division** → When forming the divisor at each step, you must double the current quotient, not the previous divisor. In step 2 of Example 2, students sometimes use 16 again instead of recalculating.

5. **Confusing square root with half** → √4 = 2, not 4/2. √9 = 3, not 4.5. The square root is the number that, when multiplied by itself, gives the original—not division by 2.

Quick Reference

• Squares 1–20 must be instant recall: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400.
• Perfect squares never end in 2, 3, 7, or 8.
• Prime factorisation method: pair up identical primes; if all pair, extract one from each pair.
• Long division: group pairs from decimal, find largest square ≤ first group, double quotient for next divisor.
• Sum of first n odd numbers = n².
• √(a×b) = √a × √b; use to simplify: √50 = √(25×2) = 5√2.