Square Root: Notes और MCQ Practice

आसान भाषा में

Square Root किसी number का वो मूल factor होता है, जिसे खुद से ही multiply करने पर original number मिल जाता है। इसे radical sign '√' या power ½ से represent करते हैं (जैसे √36 = 6, क्योंकि 6 × 6 = 36)। RRB Paramedical exam में fast calculation के लिए कम से कम 1 से 30 तक के squares याद रखना compulsory है!

Important Rules & Properties

• √(a × b) = √a × √b
• √(a ÷ b) = √a ÷ √b
• (√a)² = a
• Unit Digit Rule: अगर किसी number के end में 2, 3, 7 या 8 आए, तो वो कभी भी Perfect Square नहीं हो सकता।
• Perfect Square के end में odd number of zeroes (जैसे 1, 3, 5 zeroes) कभी नहीं होते।

2 Quick Tricks

1. Ending with 25 Rule:

अगर number 25 पर खत्म हो (e.g., √2025):

• Last digit हमेशा 5 होगी।
• बाकी बचे digits (20) को consecutive numbers के product में तोड़ें: 4 × 5 = 20.
• छोटा number चुनें (4) → Answer = 45.

1. Approximation Trick for Non-Perfect Square:

Formula: √(x ± y) ≈ √x ± [y ÷ (2√x)]

• Example for √10: इसे √(9 + 1) लिखें → √9 + [1 ÷ (2 × √9)] = 3 + 1/6 ≈ 3.16.

2 Common Mistakes

1. Addition Trap: √(a + b) कभी भी √a + √b के बराबर नहीं होता!

• Example: √(9 + 16) = √25 = 5 (इसे 3 + 4 = 7 लिखना गलत है)।

1. Decimal Error: Square Root निकालते समय decimal places की संख्या हमेशा आधी (half) हो जाती है।

• Example: √0.0064 = 0.08 होता है, 0.8 नहीं।

Square Root - Study Notes for RRB Paramedical

1. Concept

A square root of a number is a value that, when multiplied by itself, gives the original number.

• If a × a = b, then the square root of b is a.
• It is denoted by the radical symbol: √.
• Example: Since 6 × 6 = 36, we write √36 = 6.

A perfect square is an integer whose square root is also a whole number (e.g., 1, 4, 9, 16, 25, 36, 49, 64, 81, 100).

In competitive exams like RRB Paramedical, questions test your ability to quickly find square roots, simplify surds, handle decimals, and solve equation-based square root problems.

2. Important Formulas / Rules / Facts

• Multiplication Rule: √(a × b) = √a × √b
• Division Rule: √(a ÷ b) = √a ÷ √b
• Power Rule: (√a)² = a
• Addition/Subtraction Warning: √(a + b) is NOT equal to √a + √b, and √(a − b) is NOT equal to √a − √b.
• Unit Digit Pattern of Perfect Squares:
• If a number ends in 1, its square root ends in 1 or 9.
• If a number ends in 4, its square root ends in 2 or 8.
• If a number ends in 5, its square root ends in 5.
• If a number ends in 6, its square root ends in 4 or 6.
• If a number ends in 9, its square root ends in 3 or 7.
• If a number ends in 00, its square root ends in 0.
• Non-Square Rule: A perfect square never ends in the digits 2, 3, 7, or 8.
• Number of Digits in Square Root:
• If a perfect square has n digits:
• If n is even, its square root has n ÷ 2 digits.
• If n is odd, its square root has (n + 1) ÷ 2 digits.
• Decimal Rule: If a perfect square decimal has 2m decimal places, its square root will have m decimal places (half the count).

3. Solved Examples

Example 1: Finding Square Root by Prime Factorization

Question: Find the square root of 1764.

Solution:

• Step 1: Find the prime factors of 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
• Step 2: Make pairs of identical prime factors:
• Pairs = (2 × 2) × (3 × 3) × (7 × 7)
• Step 3: Take one factor from each pair and multiply them together:
• √1764 = 2 × 3 × 7 = 42

Answer: 42

Example 2: Simplifying Expressions with Decimals

Question: Evaluate √[(0.0064 × 0.25) ÷ 0.0016].

Solution:

• Step 1: Simplify the numbers inside the square root first.
• Notice that 0.0064 ÷ 0.0016 = 4 (both have 4 decimal places).
• Step 2: Rewrite the expression:
• √[4 × 0.25]
• Step 3: Multiply the terms inside:
• 4 × 0.25 = 1.00 = 1
• Step 4: Take the square root:
• √1 = 1

Answer: 1

Example 3: Finding an Unknown Variable

Question: If √(1 + x ÷ 144) = 13 ÷ 12, find the value of x.

Solution:

• Step 1: Square both sides to eliminate the square root:
• [√(1 + x ÷ 144)]² = (13 ÷ 12)²
• 1 + x ÷ 144 = 169 ÷ 144
• Step 2: Subtract 1 from both sides:
• x ÷ 144 = (169 ÷ 144) − 1
• x ÷ 144 = (169 − 144) ÷ 144
• x ÷ 144 = 25 ÷ 144
• Step 3: Multiply both sides by 144:
• x = 25

Answer: 25

Example 4: Making a Number a Perfect Square

Question: What is the smallest number by which 180 must be multiplied so that the product becomes a perfect square?

Solution:

• Step 1: Find the prime factorization of 180.
• 180 = 2 × 2 × 3 × 3 × 5
• Step 2: Group factors into pairs:
• (2 × 2) × (3 × 3) × 5
• Step 3: Identify the unpaired factor:
• 5 does not have a pair.
• Step 4: Multiply by 5 to complete the pair:
• 180 × 5 = 900, which is a perfect square (30² = 900).

Answer: 5

4. Shortcuts and Tricks

Trick 1: Fast Square Root of 4-digit Perfect Squares (e.g., √7056)

1. Look at the last digit (6): The answer must end in 4 or 6.
2. Strike out the last two digits: Discard '56'; look at the remaining number 70.
3. Find the largest square ≤ 70:

• 8² = 64 and 9² = 81. Since 64 ≤ 70, the tens digit is 8.
• The answer is either 84 or 86.

1. Decide between the two:

• Multiply the tens digit (8) by its next integer (9): 8 × 9 = 72.
• Compare the remaining number (70) with 72.
• Since 70 < 72, choose the smaller option: 84.
• Therefore, √7056 = 84.

Trick 2: Square Root of Repeated Infinite Series

• For expressions like: √(k + √(k + √(k + ... ∞)))
• If k can be written as n × (n + 1) (product of two consecutive integers):
• The answer for the '+' series is the larger number (n + 1).
• Example: √(12 + √(12 + √(12 + ... ∞))) → Since 12 = 3 × 4, answer is 4.
• The answer for the '−' series is the smaller number (n).
• Example: √(12 − √(12 − √(12 − ... ∞))) → Answer is 3.
• For √(k × √(k × √(k ... ∞))) (multiplication up to infinity):
• The answer is simply k.
• Example: √(7 × √(7 × √(7 ... ∞))) = 7.

5. Common Mistakes to Avoid

1. Confusing Decimals:

• Wrong: √0.4 = 0.2
• Correct: 0.2 × 0.2 = 0.04, so √0.04 = 0.2. However, √0.4 ≈ 0.632. Never guess decimal square roots without pairing digits after the decimal point.

1. Splitting Addition/Subtraction:

• Wrong: √(25 + 9) = √25 + √9 = 5 + 3 = 8
• Correct: √(25 + 9) = √34 ≈ 5.83. Never break a square root over '+' or '−'.

1. Ignoring Trailing Zeros:

• A number with an odd count of trailing zeros (like 40, 9000, 1600000) can never be a perfect square. A perfect square must have an even number of trailing zeros.

1. Sign Confusion:

• In basic arithmetic and government exam simplification questions, the radical sign √x denotes only the principal (non-negative) root. √25 = 5 (not −5).