Simplifying Radicals

What is a Radical?

A radical is an expression that includes a root symbol (√).

Parts of a radical:

• Radical symbol: √
• Radicand: the number/expression under the radical
• Index: the small number indicating which root (if no number shown, it's 2 for square root)

Examples:

• √25: square root of 25
• ³√8: cube root of 8
• ⁴√16: fourth root of 16

Perfect Squares

Perfect squares are numbers whose square root is a whole number.

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! They're essential for simplifying radicals.

Simplifying Square Roots

A square root is simplified when the radicand has no perfect square factors (other than 1).

Strategy:

1. Find the largest perfect square factor of the radicand
2. Rewrite as product: √(perfect square × other factor)
3. Take the square root of the perfect square
4. Leave the other factor under the radical

Example 1: √12

Find perfect square factors of 12: 12 = 4 × 3 (4 is a perfect square!)

Rewrite: √12 = √(4 × 3)

Simplify: √12 = √4 × √3 √12 = 2√3

Final answer: 2√3

Example 2: √50

Factor: 50 = 25 × 2 (25 is perfect square)

Simplify: √50 = √(25 × 2) √50 = √25 × √2 √50 = 5√2

Example 3: √72

Factor: 72 = 36 × 2 (36 is perfect square)

Simplify: √72 = √(36 × 2) √72 = 6√2

Alternative factoring: 72 = 4 × 18 √72 = 2√18

But √18 can be simplified further! 18 = 9 × 2 √18 = 3√2

So √72 = 2 × 3√2 = 6√2 ✓

Always use the LARGEST perfect square factor!

Example 4: √48

Factor: 48 = 16 × 3

Simplify: √48 = √(16 × 3) √48 = 4√3

Prime Factorization Method

For larger numbers, use prime factorization.

Example: √180

Prime factorization: 180 = 2 × 2 × 3 × 3 × 5 180 = 2² × 3² × 5

Group pairs: √180 = √(2² × 3² × 5)

Take out pairs: √180 = 2 × 3 × √5 √180 = 6√5

When Radicals Don't Simplify

Example: √15

15 = 3 × 5 (no perfect square factors)

√15 is already simplified!

Example: √7

7 is prime, no perfect square factors.

√7 is already simplified!

Simplifying Radicals with Variables

Use the same process with variables!

Rule: √(x²) = x (when x ≥ 0)

Example 1: √(x⁴)

x⁴ = (x²)²

√(x⁴) = x²

Example 2: √(x⁶)

x⁶ = (x³)²

√(x⁶) = x³

Example 3: √(16x²)

√(16x²) = √16 × √(x²) √(16x²) = 4x

Simplifying with Variables and Numbers

Example 1: √(25x⁴)

√(25x⁴) = √25 × √(x⁴) √(25x⁴) = 5x²

Example 2: √(50x⁶)

50 = 25 × 2 x⁶ = (x³)²

√(50x⁶) = √(25 × 2 × (x³)²) √(50x⁶) = 5x³√2

Example 3: √(12x⁵)

12 = 4 × 3 x⁵ = x⁴ × x = (x²)² × x

√(12x⁵) = √(4 × 3 × x⁴ × x) √(12x⁵) = 2x²√(3x)

Adding and Subtracting Radicals

You can only combine like radicals (same radicand).

Think of them like variables: 3x + 5x = 8x

Example 1: 3√2 + 5√2

Same radicand (2), so combine: 3√2 + 5√2 = 8√2

Example 2: 7√5 - 2√5

7√5 - 2√5 = 5√5

Example 3: √3 + √7

Different radicands, CANNOT combine. Answer: √3 + √7

Simplify Before Adding

Sometimes you must simplify first to see like radicals.

Example 1: √12 + √27

Simplify each: √12 = 2√3 √27 = √(9 × 3) = 3√3

Now add: 2√3 + 3√3 = 5√3

Example 2: √50 - √8

Simplify: √50 = 5√2 √8 = √(4 × 2) = 2√2

Subtract: 5√2 - 2√2 = 3√2

Example 3: 2√18 + √32

Simplify: 2√18 = 2 × 3√2 = 6√2 √32 = √(16 × 2) = 4√2

Add: 6√2 + 4√2 = 10√2

Multiplying Radicals

Rule: √a × √b = √(a × b)

Example 1: √3 × √5

√3 × √5 = √15

Example 2: √2 × √8

√2 × √8 = √16 = 4

Example 3: 2√3 × 5√2

Multiply coefficients and radicals separately: (2 × 5)(√3 × √2) = 10√6

Example 4: √6 × √6

√6 × √6 = √36 = 6

General rule: √a × √a = a

Multiplying with Simplification

Example 1: √2 × √18

√2 × √18 = √36 = 6

Example 2: √5 × √20

√5 × √20 = √100 = 10

Example 3: 3√2 × 4√8

3√2 × 4√8 = 12√16 = 12 × 4 = 48

Example 4: 2√6 × 5√3

2√6 × 5√3 = 10√18 = 10 × 3√2 = 30√2

Dividing Radicals

Rule: √a / √b = √(a/b)

Example 1: √30 / √6

√30 / √6 = √(30/6) = √5

Example 2: √50 / √2

√50 / √2 = √(50/2) = √25 = 5

Example 3: 10√15 / 2√3

Divide coefficients and radicals: (10/2)(√15/√3) = 5√5

Rationalizing the Denominator

We don't leave radicals in the denominator!

Process: Multiply numerator and denominator by the radical in the denominator.

Example 1: 3 / √2

Multiply by √2/√2: (3 × √2) / (√2 × √2) = 3√2 / 2

Example 2: 5 / √3

Multiply by √3/√3: (5 × √3) / (√3 × √3) = 5√3 / 3

Example 3: 8 / √8

First simplify √8 = 2√2: 8 / 2√2 = 4 / √2

Rationalize: (4 × √2) / (√2 × √2) = 4√2 / 2 = 2√2

Rationalizing with Binomials

When denominator is a + √b, multiply by the conjugate a - √b.

Example: 1 / (2 + √3)

Conjugate of 2 + √3 is 2 - √3

Multiply: (1 × (2 - √3)) / ((2 + √3)(2 - √3))

Denominator: (2 + √3)(2 - √3) = 4 - 3 = 1

Result: 2 - √3

Cube Roots and Higher

Cube root: ³√a means what number cubed equals a?

Perfect cubes:

• 1³ = 1
• 2³ = 8
• 3³ = 27
• 4³ = 64
• 5³ = 125
• 6³ = 216

Example 1: ³√8 = 2 (because 2³ = 8)

Example 2: ³√27 = 3

Example 3: ³√64 = 4

Example 4: ³√125 = 5

Simplifying Cube Roots

Example 1: ³√24

Factor: 24 = 8 × 3 (8 is perfect cube)

³√24 = ³√(8 × 3) = ³√8 × ³√3 = 2³√3

Example 2: ³√54

54 = 27 × 2

³√54 = ³√(27 × 2) = 3³√2

Product Rule for Radicals

√a × √b = √(ab) works for any index.

Example (cube roots): ³√2 × ³√4

³√2 × ³√4 = ³√8 = 2

Radicals in Equations

Example: Solve √x = 5

Square both sides: (√x)² = 5² x = 25

Check: √25 = 5 ✓

Example 2: Solve √(x + 3) = 7

Square both sides: x + 3 = 49 x = 46

Check: √(46 + 3) = √49 = 7 ✓

Applications: Pythagorean Theorem

For right triangles: a² + b² = c²

Example: Legs are 3 and 4. Find hypotenuse.

3² + 4² = c² 9 + 16 = c² 25 = c² c = √25 = 5

Example 2: Hypotenuse is 10, one leg is 6. Find other leg.

6² + b² = 10² 36 + b² = 100 b² = 64 b = √64 = 8

Applications: Area and Geometry

Example: Area of square is 50 cm². Find side length.

Side = √50 = √(25 × 2) = 5√2 cm

Approximate: 5 × 1.414 ≈ 7.07 cm

Common Mistakes to Avoid

1. √(a + b) ≠ √a + √b √(9 + 16) = √25 = 5 NOT √9 + √16 = 3 + 4 = 7

2. Not using largest perfect square Use √(36 × 2) not √(4 × 18) for √72

3. Adding unlike radicals √2 + √3 cannot be simplified!

4. Forgetting to simplify final answer Leave answer as 2√3, not √12

5. Losing negative signs -5√2 + 3√2 = -2√2, not 2√2

6. Not rationalizing denominators Final answer should not have √ in denominator

Quick Reference

Simplifying: Find perfect square factor, take it out

Adding/Subtracting: Only combine like radicals

Multiplying: √a × √b = √(ab)

Dividing: √a / √b = √(a/b)

Rationalizing: Multiply by √n/√n to remove √ from denominator

Practice Strategy

Level 1: Perfect squares

• √4, √16, √36, √100

Level 2: Simple simplification

• √12, √18, √20, √50

Level 3: Larger numbers

• √72, √98, √200

Level 4: With variables

• √(x⁴), √(25x²), √(18x⁶)

Level 5: Operations

• Add, subtract, multiply, divide radicals

Tips for Success

• Memorize perfect squares 1-225
• Look for largest perfect square factor
• Simplify before adding or subtracting
• Always rationalize denominators
• Check work by squaring result
• Practice prime factorization
• Draw factor trees for complex numbers
• Remember √a × √a = a
• Keep track of coefficients separately
• Simplify your final answer completely