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Conjugate Pairs

Part of the Radicals & Rational Exponents section of Coddy's Algebra 2 journey. Lesson 14 of 43.

Multiply (4 - √(3))(4 + √(3)). The four products are 16, 4√(3), -4√(3) and -3. The two middle terms are opposites and cancel, so what is left is 16 minus 3, which is 13. No root survives.

This always happens. The brackets a - √(b) and a + √(b) differ only in the sign between their terms; they are called conjugates. Their product is a difference of squares:

(a - √(b))(a + √(b)) = a^2 - b

That is the reason conjugates matter: when a and b are whole numbers, they turn an expression with a root into a whole number. In the next chapter this is how a root is cleared from the bottom of a fraction.

Challenge

Expand and simplify (5 - √(2))(5 + √(2)).

Try it yourself

(5 - sqrt(2))(5 + sqrt(2))

Write it in simplest radical form

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Radicals & Rational Exponents

1Simplifying Radicals

Roots and Powers Perfect Square Factors The Largest Square Letters Under the Root Cube Roots Recap: Simplifying Radicals

4Rationalizing

One Root Below A Number and a Root The Conjugate Below Recap: Rationalizing

7Powers and Roots Equations

Odd Powers Even Powers Fractional Powers Two Thirds Power Recap: Power Equations

2Adding & Subtracting Radicals

Like Radicals Simplify First Subtracting Recap: Adding Radicals

5Rational Exponents

One Over n m Over n Negative Exponents Exponent Rules Still Work Recap: Rational Exponents

8Radical Functions

Domain of a Root Evaluating Inverse of a Square Recap: Radical Functions

3Multiplying Radicals

Root Times Root Distributing a Root Two BracketsConjugate PairsRecap: Multiplying Radicals

6Radical Equations

Square Both Sides Isolate the Root First A False Answer Cube Root Equations Recap: Radical Equations

9Final Challenges

Final: Simplify and Add Final: Clear the Root Final: The Exponent Final: Check It Final: The Falling Ball