Unit 3b: Complex Numbers

Lesson 3.2b Conjugates with Radicals

Introduction

02.10.26

Conjugate radicals or binomial pairs like 𝐚+√𝐛 and 𝐚-√𝐛 that share the same terms, but have opposite signs between them. Their product always eliminates the radical, resulting in an integer.

(𝐚+√𝐛)(𝐚-√𝐛)=𝐚²-𝐛

*The goal is not to eliminate a radical completely but as a step in the solving process, leading us towards an asemtope unit

1. multiply the entire fraction by the conjugate of whichever part of the fraction contains a radical expression.

            ⭐T️he conjugate is essentially the inverse. Because we can’t do anything with the                 actual radical, only the other signs like the added or subtracted piece are changed.

2. Do not cross multiply! (Unlike geometry) Only multiply straight across. Because the middle terms of multiplication cancel out in the denominator, the square root is simply undone and the rest of multiplied as usual.

            ⭐️Pay attention to the different signs of the conjugates in multiplication

3. Do not solve/double distribute the actual top part (we don’t have to as of now) but instead wait until the final simplification to see if there is a GCF, then divide everything by that to simplify.

Examples with 𝐢

• Similar to dealing with radicals

Tips for Solving Conjugate Radicals with Imaginary Numbers

Algebra II (Accelerated) – Test Prep

1. Know why you use conjugates

You multiply by a conjugate to:

• Eliminate imaginary terms in the denominator

• Eliminate radicals when they’re paired with an imaginary number

Key idea:
(𝐚 + 𝐛𝐢)(𝐚 − 𝐛𝐢) = 𝐚² + 𝐛²

The middle terms cancel because 𝐢² = −1.

2. Always change negative radicals first

Before doing anything else, rewrite square roots of negatives:

√(−𝐚) = 𝐢√𝐚

Example:
√(−18) = 𝐢√18 = 3𝐢√2

This step is often skipped — and it’s a common test mistake.

3. Conjugates: change the sign only in the middle

If you see:

• 𝐚 + 𝐛𝐢√𝐚 → conjugate is 𝐚 − 𝐛𝐢√𝐚

• 𝐢√5 − 3 → conjugate is 𝐢√5 + 3

Do not change:

• exponents

• coefficients

• radicals themselves

4. Multiply entire numerator and denominator

When rationalizing denominators:

• Multiply the entire fraction by the conjugate

• FOIL carefully

Example:
2 / (3 + 𝐢√5) · (3 − 𝐢√5) / (3 − 𝐢√5)

5. Memorize this shortcut (huge time-saver)

When multiplying conjugates:

(𝐚 + 𝐛𝐢)(𝐚 − 𝐛𝐢) = 𝐚² + 𝐛²

Example:
(4 + 3𝐢)(4 − 3𝐢) = 16 + 9 = 25

No FOIL needed if it’s a true conjugate pair.

6. Watch the sign of 𝐢²

Always replace:

𝐢² = −1

Example:
(𝐢√7)² = −7

Many students lose points by writing +7.

7. Final answers should look like this

Depending on the problem:

• No imaginary numbers in the denominator

• Simplified radicals

• Written in standard form like 𝐚 + 𝐛𝐢 when possible

8. Common test traps to avoid

• Forgetting to multiply both numerator and denominator

• Leaving √(−𝐚) instead of converting to 𝐢√𝐚

• Sign errors with 𝐢²

• Not simplifying radicals at the end

9. What to practice the night before

Focus on:

• Rationalizing denominators with 𝐢

• Multiplying conjugates

• Writing answers in standard form 𝐚 + 𝐛𝐢