📌 Introduction
Square Roots and Surds are essential for simplifying expressions, solving equations, and dealing with irrational numbers. You’ll often find these in arithmetic simplification, algebraic identities, and even Data Interpretation-based questions.
📌 Definitions
🔹 Square Root (√):
The square root of a number x is a number y such that y² = x.
Example: √25 = 5, because 5² = 25.
Note: Square roots of perfect squares (like 4, 9, 16, 25) are rational. Others are irrational.
🔹 Surd:
A surd is an irrational root of a rational number. It cannot be simplified to remove the root sign completely.
Examples:
√2, √3, √5 are surds
√4 = 2 is not a surd (because it simplifies to a rational number)
📌 Important Properties of Square Roots
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√(a × b) = √a × √b
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√(a / b) = √a / √b
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(√a)² = a
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√a + √b ≠ √(a + b)
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(a + √b)(a - √b) = a² - b
📌 Common Square Roots to Remember
1² = 1 6² = 36 11² = 121
2² = 4 7² = 49 12² = 144
3² = 9 8² = 64 13² = 169
4² = 16 9² = 81 14² = 196
5² = 25 10² = 100 15² = 225
📌 Simplification of Surds
To simplify:
√72 = √(36 × 2) = √36 × √2 = 6√2
General rule: Factor the number into squares and simplify the square roots.
📌 Rationalizing the Denominator
Rationalizing means converting a denominator from irrational to rational by multiplying with a suitable form.
Examples:
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1 / √2 = (1 / √2) × (√2 / √2) = √2 / 2
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1 / (2 + √3) → multiply numerator & denominator by (2 - √3):
(1 / (2 + √3)) × ((2 - √3) / (2 - √3))
= (2 - √3) / (4 - 3) = (2 - √3) / 1 = 2 - √3
📌 Surd Laws (Useful Shortcuts)
Let a, b be positive rational numbers and m, n be integers:
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√a × √b = √(ab)
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√a / √b = √(a/b)
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(a + √b)(a - √b) = a² - b
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1 / (a + √b) = (a - √b) / (a² - b)
📌 Approximating Square Roots
For non-perfect squares:
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√2 ≈ 1.414
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√3 ≈ 1.732
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√5 ≈ 2.236
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√10 ≈ 3.162
Use these values for approximation problems in exams.
📌 Important Patterns
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(√a + √b)² = a + b + 2√ab
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(√a - √b)² = a + b - 2√ab
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(a + √b)(a - √b) = a² - b
📌 Practice Examples
Example 1: Simplify √50 + √18
= √(25 × 2) + √(9 × 2)
= 5√2 + 3√2 = 8√2
Example 2: Rationalize 1 / (√5 - √3)
Multiply by (√5 + √3):
= (√5 + √3) / (5 - 3) = (√5 + √3) / 2
Example 3: If √x = 5, find x.
⇒ x = (√x)² = 5² = 25
Example 4: Simplify (√3 + 2)²
= (√3)² + 2×2×√3 + 2² = 3 + 4√3 + 4 = 7 + 4√3
📌 Common Mistakes to Avoid
❌ √(a + b) ≠ √a + √b
❌ √(a - b) ≠ √a - √b
✅ Always factor numbers into perfect squares for simplification
✅ Always rationalize denominators if required
📌 Practice MCQs
Q1. Simplify: √12 + √27
A. 6√3
B. 3√5
C. 5√3
D. 3√3 + √3
✔️ Correct: A
√12 = 2√3, √27 = 3√3 → Total = 5√3
Q2. Rationalize: 1 / √3
A. √3 / 2
B. √3
C. √3 / 3
D. 1 / 3
✔️ Correct: C
Multiply by √3 ⇒ √3 / 3
Q3. Find x if √x = 7
A. 14
B. 49
C. 21
D. 13
✔️ Correct: B
x = (√x)² = 49
Q4. (√7 - 2)² = ?
A. 11 - 4√7
B. 7 - 4√7 + 4
C. 11 + 4√7
D. 7 + 4
✔️ Correct: A
= 7 - 4√7 + 4 = 11 - 4√7
📌 Summary Tips
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Memorize squares from 1 to 30 for speed.
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Simplify surds using factorization.
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Rationalize when the denominator has a surd.
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Apply algebraic identities involving roots.
