Operations with radicals

Adding and subtracting

Only like radicals (same index, same radicand) can be combined, just like like terms.

• 3√5 + 7√5 = 10√5
• 8√2 − 3√2 = 5√2
• √3 + √5 can’t be combined.

Simplify first: radicals that look different may be alike.

Example: √12 + √27 = 2√3 + 3√3 = 5√3

Multiplying

Product rule: √a × √b = √(ab)

Multiply outside numbers together and radicands together.

• √3 × √12 = √36 = 6
• 2√5 × 3√2 = 6√10
• √6 × √10 = √60 = 2√15

Distributing: √2(√8 + 3) = √16 + 3√2 = 4 + 3√2

Binomials: (√3 + 2)(√3 − 5) = 3 − 5√3 + 2√3 − 10 = −7 − 3√3

Dividing

Quotient rule: √a ÷ √b = √(a/b)

• √50 ÷ √2 = √25 = 5
• 12√15 ÷ 4√3 = 3√5

Rationalizing the denominator

By convention, simplified answers don’t have radicals in the denominator. To rationalize the denominator, multiply the top and bottom by the radical.

Example: 6/√3

• Multiply by √3/√3: 6√3/3 = 2√3

Binomial denominators: multiply by the conjugate.

Example: 4/(√5 + 1)

• × (√5 − 1)/(√5 − 1)
• Denominator: 5 − 1 = 4
• Result: 4(√5 − 1)/4 = √5 − 1

The conjugate works because (a + b)(a − b) = a² − b² removes the radical.

Summary table

Worked examples

Example 1: 5√7 − 2√7 + √7 → 4√7.

Example 2: √8 × √18 → √144 = 12.

Example 3: 10/√5 → 10√5/5 = 2√5.

Common mistakes

• Adding unlike radicals: √2 + √3 ≠ √5.
• √a + √b = √(a + b): false.
• Forgetting to simplify the final radical.
• Only multiplying the denominator when rationalizing.

Try it yourself

1. 4√3 + √48
2. √5 × √20
3. 3√2(√6 − 1)
4. 9/√3
5. 2/(√3 − 1)

Answers: 1) 8√3 2) 10 3) 6√3 − 3√2 4) 3√3 5) √3 + 1

Quick recap

• Combine only like radicals.
• Use the product rule and quotient rule.
• Rationalize the denominator with the radical or the conjugate.