Adding and Subtracting Radicals

Combining like radicals

Radical Operations

Review: Simplifying Radicals

Before performing operations, always simplify radicals first.

Key reminders:

Find perfect square factors
√(a²) = a
√(ab) = √a × √b
Simplify coefficients and radicands separately

Quick examples:

√50 = 5√2
√75 = 5√3
√12 = 2√3

Adding and Subtracting Radicals

Rule: You can ONLY add or subtract like radicals (same radicand).

Think of radicals like variables:

3x + 5x = 8x
3√2 + 5√2 = 8√2

Example 1: 4√3 + 7√3

Same radicand (3), so combine coefficients: 4√3 + 7√3 = 11√3

Example 2: 9√5 - 2√5

9√5 - 2√5 = 7√5

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

Adding unlike radicals √2 + √3 ≠ √5 (cannot combine!)
Distributing exponents incorrectly (√a + √b)² ≠ a + b Must FOIL: (√a + √b)² = a + 2√(ab) + b
Forgetting to simplify Leave 2√3, not √12
Rationalizing errors Don't forget to multiply BOTH numerator and denominator
Sign errors with conjugates (a + √b)(a - √b) = a² - b, not a² + b
Not combining like terms 5√2 - 3√2 = 2√2, not 5√2 - 3√2

Order of Operations with Radicals

Follow PEMDAS/GEMDAS:

Parentheses/Grouping
Exponents (including radicals)
Multiplication and Division (left to right)
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

📚 Practice Problems

1Problem 1easy

❓ Question:

Simplify: 5√3 + 2√3

💡 Show Solution

Step 1: Check if the radicals are like radicals: Both terms have √3, so they are like radicals.

Step 2: Add the coefficients: Just like adding 5x + 2x = 7x 5√3 + 2√3 = (5 + 2)√3

Step 3: Simplify: = 7√3

Think of √3 as the "unit" you're counting.

Answer: 7√3

2Problem 2easy

❓ Question:

Add: $5\sqrt{7} + 3\sqrt{7}$

💡 Show Solution

These are like radicals (both have $\sqrt{7}$ ).

Combine the coefficients: $5\sqrt{7} + 3\sqrt{7} = (5 + 3)\sqrt{7} = 8\sqrt{7}$

Answer: $8\sqrt{7}$

3Problem 3easy

❓ Question:

Simplify: 8√5 - 3√5

💡 Show Solution

Step 1: Check if the radicals are like radicals: Both terms have √5, so they are like radicals.

Step 2: Subtract the coefficients: Just like 8x - 3x = 5x 8√5 - 3√5 = (8 - 3)√5

Step 3: Simplify: = 5√5

Answer: 5√5

4Problem 4medium

❓ Question:

Simplify: $\sqrt{18} + \sqrt{8}$

💡 Show Solution

Step 1: Simplify each radical $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$ $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$

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

Answer: $5\sqrt{2}$

5Problem 5medium

❓ Question:

Simplify: 3√2 + 4√3 - √2

💡 Show Solution

Step 1: Identify like radicals: Terms with √2: 3√2 and -√2 Terms with √3: 4√3 (only one, cannot be combined)

Step 2: Combine like radicals: 3√2 - √2 = (3 - 1)√2 = 2√2 (Note: √2 is the same as 1√2)

Step 3: Write the final answer: 2√2 + 4√3

This cannot be simplified further because √2 and √3 are not like radicals.

Answer: 2√2 + 4√3

6Problem 6medium

❓ Question:

Simplify: √12 + √27

💡 Show Solution

Step 1: Simplify each radical first:

√12 = √(4 × 3) = √4 × √3 = 2√3

√27 = √(9 × 3) = √9 × √3 = 3√3

Step 2: Rewrite the expression with simplified radicals: 2√3 + 3√3

Step 3: Now they are like radicals! Add the coefficients: (2 + 3)√3 = 5√3

Important: Always simplify radicals first before trying to combine them.

Answer: 5√3

7Problem 7medium

❓ Question:

Multiply: $\sqrt{6} \cdot \sqrt{10}$

💡 Show Solution

Use the product property: $\sqrt{6} \cdot \sqrt{10} = \sqrt{6 \cdot 10} = \sqrt{60}$

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

Answer: $2\sqrt{15}$

8Problem 8hard

❓ Question:

Simplify: 2√50 - 3√8 + √32

💡 Show Solution

Step 1: Simplify each radical:

2√50 = 2√(25 × 2) = 2 × 5√2 = 10√2

3√8 = 3√(4 × 2) = 3 × 2√2 = 6√2

√32 = √(16 × 2) = 4√2

Step 2: Rewrite with simplified radicals: 10√2 - 6√2 + 4√2

Step 3: All are like radicals! Combine coefficients: (10 - 6 + 4)√2 = 8√2

Step 4: Verify each simplification: 2√50 = 2√(25·2) = 10√2 ✓ 3√8 = 3√(4·2) = 6√2 ✓ √32 = √(16·2) = 4√2 ✓

Answer: 8√2

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