Introduction to Radicals

What is a Square Root?

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

Symbol: √ (radical sign)

Example: √16 = 4

• Because 4 × 4 = 16

Read as: "The square root of 16 equals 4"

Perfect Squares

Perfect square: A number that has a whole number square root

First 15 perfect squares:

• 1² = 1
• 2² = 4
• 3² = 9
• 4² = 16
• 5² = 25
• 6² = 36
• 7² = 49
• 8² = 64
• 9² = 81
• 10² = 100
• 11² = 121
• 12² = 144
• 13² = 169
• 14² = 196
• 15² = 225

Memorize these for quick calculations!

Finding Square Roots

Example 1: Perfect Square

√49 = ?

Ask: What number times itself equals 49?

• 7 × 7 = 49

Answer: √49 = 7

Example 2: Another Perfect Square

√100 = ?

10 × 10 = 100

Answer: √100 = 10

Example 3: Small Perfect Square

√1 = 1 (because 1 × 1 = 1) √0 = 0 (because 0 × 0 = 0)

Estimating Square Roots

For non-perfect squares: Estimate between two perfect squares

Example: Estimate √50

Find perfect squares around 50:

• 49 < 50 < 64
• 7² < 50 < 8²

So: 7 < √50 < 8

Closer to: √50 ≈ 7.1 (closer to 49 than 64)

Example: Estimate √30

25 < 30 < 36 5² < 30 < 6²

So: 5 < √30 < 6

Estimate: √30 ≈ 5.5

Simplifying Radicals

Simplified form: No perfect square factors under the radical (except 1)

Method: Find largest perfect square factor

Rule: √(a × b) = √a × √b

Example 1: Simplify √12

Step 1: Find perfect square factor

• 12 = 4 × 3
• 4 is a perfect square!

Step 2: Split the radical

• √12 = √(4 × 3)
• = √4 × √3

Step 3: Simplify

• = 2√3

Answer: √12 = 2√3

Example 2: Simplify √50

Factor: 50 = 25 × 2

Simplify:

√50 = √(25 × 2)
= √25 × √2
= 5√2

Answer: 5√2

Example 3: Simplify √75

Factor: 75 = 25 × 3

Simplify:

√75 = √(25 × 3)
= √25 × √3
= 5√3

Answer: 5√3

Example 4: Already Simplified

√17

17 has no perfect square factors

Answer: √17 (already simplified)

Adding and Subtracting Radicals

Only combine like radicals (same number under radical)

Like: 3√2 and 5√2 (both have √2) Unlike: 3√2 and 5√3 (different radicals)

Example 1: Add Like Radicals

3√5 + 2√5

Same radical (√5), so add coefficients:

• (3 + 2)√5 = 5√5

Answer: 5√5

Example 2: Subtract

7√3 − 4√3

(7 − 4)√3 = 3√3

Answer: 3√3

Example 3: Simplify First

√12 + √27

Step 1: Simplify each

• √12 = 2√3
• √27 = √(9 × 3) = 3√3

Step 2: Add like radicals

• 2√3 + 3√3 = 5√3

Answer: 5√3

Example 4: Cannot Combine

2√5 + 3√2

Different radicals, cannot combine!

Answer: 2√5 + 3√2 (leave as is)

Multiplying Radicals

Rule: √a × √b = √(ab)

Example 1: Multiply

√3 × √5

= √(3 × 5) = √15

Answer: √15

Example 2: With Coefficients

2√3 × 4√5

Multiply coefficients: 2 × 4 = 8 Multiply radicals: √3 × √5 = √15

Answer: 8√15

Example 3: Same Radical

√7 × √7

= √(7 × 7) = √49 = 7

Answer: 7

General rule: √a × √a = a

Dividing Radicals

Rule: √a / √b = √(a/b)

Example 1: Divide

√20 / √5

= √(20/5) = √4 = 2

Answer: 2

Example 2: Simplify Result

√18 / √2

= √(18/2) = √9 = 3

Answer: 3

Rationalizing the Denominator

Goal: No radical in denominator

Method: Multiply top and bottom by the radical

Example: Rationalize 1/√3

Multiply by √3/√3:

1/√3 × √3/√3 = √3/3

Answer: √3/3

Product Property of Square Roots

√(a²) = a (for positive a)

Example: Simplify √(5²)

√25 = 5

Also: √(x²) = x (when x ≥ 0)

Real-World Applications

Geometry: Side length from area

• Square area = 64 cm²
• Side = √64 = 8 cm

Pythagorean Theorem: Finding hypotenuse

• a² + b² = c²
• c = √(a² + b²)

Physics: Free fall distance

• d = √(2h/g)

Engineering: Calculating loads and stresses

Practice