Square Roots and Surds

 📌 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

1. √(a × b) = √a × √b

2. √(a / b) = √a / √b

3. (√a)² = a

4. √a + √b ≠ √(a + b)

5. (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:

1. 1 / √2 = (1 / √2) × (√2 / √2) = √2 / 2

2. 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:

• ⁠√a × √b = √(ab)

• ⁠√a / √b = √(a/b)

• ⁠(a + √b)(a - √b) = a² - b

• ⁠1 / (a + √b) = (a - √b) / (a² - b)

 📌 Approximating Square Roots

For non-perfect squares:

• √2 ≈ 1.414

• √3 ≈ 1.732

• √5 ≈ 2.236

• √10 ≈ 3.162

Use these values for approximation problems in exams.

 📌 Important Patterns

• (√a + √b)² = a + b + 2√ab

• (√a - √b)² = a + b - 2√ab

• (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

• Memorize squares from 1 to 30 for speed.

• Simplify surds using factorization.

• Rationalize when the denominator has a surd.

• Apply algebraic identities involving roots.