Example 3: 6√7 + 3√7 - √7

Combine all terms: (6 + 3 - 1)√7 = 8√7

Note: √7 = 1√7

Example 4: √2 + √3

Different radicands - CANNOT combine! Answer: √2 + √3

Simplify First, Then Add

Often you must simplify radicals before you can see they're alike.

Example 1: √12 + √27

Simplify each: √12 = √(4 × 3) = 2√3 √27 = √(9 × 3) = 3√3

Now add like radicals: 2√3 + 3√3 = 5√3

Example 2: √50 + √32

Simplify: √50 = √(25 × 2) = 5√2 √32 = √(16 × 2) = 4√2

Add: 5√2 + 4√2 = 9√2

Example 3: √8 - √18 + √50

Simplify each: √8 = 2√2 √18 = 3√2 √50 = 5√2

Combine: 2√2 - 3√2 + 5√2 = 4√2

Example 4: 3√20 - √45

Simplify: 3√20 = 3 × 2√5 = 6√5 √45 = 3√5

Subtract: 6√5 - 3√5 = 3√5

Mixed Radical Expressions

Example 1: 2√3 + √12 - √27

Simplify: √12 = 2√3 √27 = 3√3

Rewrite: 2√3 + 2√3 - 3√3 = √3

Example 2: 4√8 + 2√2 - √32

Simplify: 4√8 = 4 × 2√2 = 8√2 √32 = 4√2

Combine: 8√2 + 2√2 - 4√2 = 6√2

Multiple Different Radicals

Keep unlike radicals separate.

Example: 2√3 + 5√2 - √3 + 3√2

Group like radicals: (2√3 - √3) + (5√2 + 3√2) = √3 + 8√2

Final answer: √3 + 8√2

Multiplying Radicals

Product Rule: √a × √b = √(ab)

Example 1: √2 × √3

√2 × √3 = √6

Example 2: √5 × √5

√5 × √5 = √25 = 5

General rule: √a × √a = a

Example 3: √6 × √10

√6 × √10 = √60

Simplify: √60 = √(4 × 15) = 2√15

Example 4: √3 × √12

√3 × √12 = √36 = 6

Multiplying with Coefficients

Multiply coefficients separately from radicals.

Example 1: (2√3)(5√2)

Coefficients: 2 × 5 = 10 Radicals: √3 × √2 = √6

Answer: 10√6

Example 2: (3√5)(4√5)

Coefficients: 3 × 4 = 12 Radicals: √5 × √5 = 5

Answer: 12 × 5 = 60

Example 3: (6√2)(3√8)

Coefficients: 6 × 3 = 18 Radicals: √2 × √8 = √16 = 4

Answer: 18 × 4 = 72

Example 4: (-2√7)(5√3)

Coefficients: -2 × 5 = -10 Radicals: √7 × √3 = √21

Answer: -10√21

Multiplying and Simplifying

Example 1: (2√6)(3√15)

Multiply: (2 × 3)(√6 × √15) = 6√90

Simplify √90: 90 = 9 × 10 √90 = 3√10

Answer: 6 × 3√10 = 18√10

Example 2: (5√12)(2√3)

First simplify √12 = 2√3: (5 × 2√3)(2√3) = (10√3)(2√3)

Multiply: (10 × 2)(√3 × √3) = 20 × 3 = 60

Distributive Property with Radicals

Example 1: √2(√3 + √5)

Distribute: √2 × √3 + √2 × √5 = √6 + √10

Example 2: 3√5(2√5 - 4)

Distribute: 3√5 × 2√5 - 3√5 × 4 = 6 × 5 - 12√5 = 30 - 12√5

Example 3: 2√3(√12 + √27)

First simplify: √12 = 2√3 √27 = 3√3

Distribute: 2√3(2√3 + 3√3) = 2√3 × 2√3 + 2√3 × 3√3 = 4 × 3 + 6 × 3 = 12 + 18 = 30

Multiplying Binomials with Radicals (FOIL)

Use FOIL: First, Outer, Inner, Last

Example 1: (√3 + 2)(√3 + 5)

F: √3 × √3 = 3 O: √3 × 5 = 5√3 I: 2 × √3 = 2√3 L: 2 × 5 = 10

Combine: 3 + 5√3 + 2√3 + 10 = 13 + 7√3

Example 2: (√5 + 1)(√5 - 1)

F: √5 × √5 = 5 O: √5 × (-1) = -√5 I: 1 × √5 = √5 L: 1 × (-1) = -1

Combine: 5 - √5 + √5 - 1 = 4

Notice: This is the difference of squares pattern!

Example 3: (2√3 + 4)(√3 - 2)

F: 2√3 × √3 = 2 × 3 = 6 O: 2√3 × (-2) = -4√3 I: 4 × √3 = 4√3 L: 4 × (-2) = -8

Combine: 6 - 4√3 + 4√3 - 8 = -2

Conjugates

Conjugates differ only in the sign between terms:

• a + √b and a - √b are conjugates
• √a + √b and √a - √b are conjugates

When you multiply conjugates, the radicals disappear!

Pattern: (a + √b)(a - √b) = a² - b

Example 1: (3 + √2)(3 - √2)

= 9 - 2 = 7

Example 2: (√5 + √3)(√5 - √3)

= 5 - 3 = 2

Example 3: (2√7 + 1)(2√7 - 1)

= (2√7)² - 1² = 4 × 7 - 1 = 28 - 1 = 27

Dividing Radicals

Quotient Rule: √a / √b = √(a/b)

Example 1: √30 / √6

= √(30/6) = √5

Example 2: √72 / √8

= √(72/8) = √9 = 3

Example 3: √50 / √2

= √(50/2) = √25 = 5

Dividing with Coefficients

Divide coefficients and radicals separately.

Example 1: 12√10 / 3√2

Coefficients: 12/3 = 4 Radicals: √10/√2 = √5

Answer: 4√5

Example 2: 20√15 / 5√3

= (20/5)(√15/√3) = 4√5

Example 3: 18√24 / 6√3

First simplify √24 = 2√6: = 18 × 2√6 / 6√3 = 36√6 / 6√3 = 6(√6/√3) = 6√2

Rationalizing the Denominator

Never leave a radical in the denominator!

Method: Multiply by a form of 1 that eliminates the radical.

Example 1: 1/√3

Multiply by √3/√3: = (1 × √3)/(√3 × √3) = √3/3

Example 2: 5/√2

= (5√2)/(√2 × √2) = 5√2/2

Example 3: 8/√8

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

Rationalize: = 4√2/2 = 2√2

Rationalizing with Variables

Example 1: 3/√x

= (3√x)/(√x × √x) = 3√x/x

Example 2: 5/(2√y)

= (5√y)/(2√y × √y) = 5√y/(2y)

Rationalizing Binomial Denominators

Use the conjugate to rationalize.

Example 1: 6/(2 + √3)

Multiply by conjugate (2 - √3)/(2 - √3):

= 6(2 - √3)/[(2 + √3)(2 - √3)] = 6(2 - √3)/(4 - 3) = 6(2 - √3)/1 = 12 - 6√3

Example 2: 10/(√5 + 1)

Conjugate: (√5 - 1)/(√5 - 1)

= 10(√5 - 1)/[(√5 + 1)(√5 - 1)] = 10(√5 - 1)/(5 - 1) = 10(√5 - 1)/4 = 5(√5 - 1)/2 = (5√5 - 5)/2

Example 3: 1/(3 - √2)

Multiply by (3 + √2)/(3 + √2):

= (3 + √2)/[(3 - √2)(3 + √2)] = (3 + √2)/(9 - 2) = (3 + √2)/7

Complex Rationalization

Example: (√3 + 1)/(√3 - 1)

Multiply by (√3 + 1)/(√3 + 1):

= (√3 + 1)²/[(√3 - 1)(√3 + 1)] = (3 + 2√3 + 1)/(3 - 1) = (4 + 2√3)/2 = 2 + √3

Simplifying Complex Radical Expressions

Example 1: (2 + √12)/4

First simplify √12 = 2√3: = (2 + 2√3)/4

Factor numerator: = 2(1 + √3)/4 = (1 + √3)/2

Example 2: (6 - √18)/3

Simplify √18 = 3√2: = (6 - 3√2)/3

Factor: = 3(2 - √2)/3 = 2 - √2

Higher Index Radicals

Cube roots: ³√a × ³√b = ³√(ab)

Example 1: ³√2 × ³√4

= ³√8 = 2

Example 2: ³√5 × ³√25

= ³√125 = 5

Example 3: ³√16 / ³√2

= ³√(16/2) = ³√8 = 2

Mixed Operations

Example 1: (√12 + √27) - (√75 - √48)

Simplify each: = (2√3 + 3√3) - (5√3 - 4√3) = 5√3 - √3 = 4√3

Example 2: 2√18 × √2 + √32

= 2√36 + √32 = 12 + 4√2

Example 3: (√50 + √8)/√2

= (5√2 + 2√2)/√2 = 7√2/√2 = 7

Applications: Pythagorean Theorem

Example: Right triangle with legs 2√3 and 4.

Find hypotenuse: c² = (2√3)² + 4² c² = 4 × 3 + 16 c² = 12 + 16 c² = 28 c = √28 = 2√7

Applications: Geometric Mean

The geometric mean of a and b is √(ab).

Example: Find geometric mean of 8 and 18.

√(8 × 18) = √144 = 12

Example: Find geometric mean of 6 and 24.

√(6 × 24) = √144 = 12

Applications: Distance and Area

Example: Square has area 50 cm². Find side length.

Side = √50 = 5√2 cm

Perimeter = 4 × 5√2 = 20√2 cm

Common Mistakes to Avoid

1. Adding unlike radicals √2 + √3 ≠ √5 (cannot combine!)

2. Distributing exponents incorrectly (√a + √b)² ≠ a + b Must FOIL: (√a + √b)² = a + 2√(ab) + b

3. Forgetting to simplify Leave 2√3, not √12

4. Rationalizing errors Don't forget to multiply BOTH numerator and denominator

5. Sign errors with conjugates (a + √b)(a - √b) = a² - b, not a² + b

6. Not combining like terms 5√2 - 3√2 = 2√2, not 5√2 - 3√2

Order of Operations with Radicals

Follow PEMDAS/GEMDAS:

1. Parentheses/Grouping
2. Exponents (including radicals)
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

Example: 2√3 + √12 × √3

Multiply first: √12 × √3 = √36 = 6 Then add: 2√3 + 6

Quick Reference

Adding/Subtracting: Only combine like radicals (same radicand)

Multiplying: √a × √b = √(ab) Multiply coefficients separately

Dividing: √a / √b = √(a/b)

Rationalizing: Multiply by √n/√n or conjugate

Conjugate pattern: (a + √b)(a - √b) = a² - b

Practice Strategy

Level 1: Add/subtract like radicals

• 3√2 + 5√2

Level 2: Simplify first, then add

• √12 + √27

Level 3: Multiply radicals

• (2√3)(5√2)

Level 4: FOIL with radicals

• (√5 + 2)(√5 - 3)

Level 5: Rationalize denominators

• 1/(2 + √3)

Tips for Success

• Always simplify radicals first
• Only combine like radicals
• Multiply coefficients separately from radicals
• Rationalize all denominators
• Use conjugates for binomial denominators
• Check your work by approximating decimal values
• Remember √a × √a = a
• Watch for conjugate patterns
• Factor when possible to simplify
• Practice FOIL with radicals until automatic

Step 2: Now they are like radicals, so add them $3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$

Answer: $5\sqrt{2}$

Now simplify: $\sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15}$

Answer: $2\sqrt{15}$