Radical Operations Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
Expand and simplify (3+2)(3−2)(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}).

Solution

  1. 1
    This is a difference of squares: (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2.
  2. 2
    (3)2−(2)2=3−2=1(\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1.

Answer

11
The conjugate product (a+b)(a−b)=a−b(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b always eliminates the radicals. This is the basis of rationalizing denominators.

About Radical Operations

Adding, subtracting, and multiplying expressions that contain radicals. Like terms (same radicand) can be combined for addition and subtraction; for multiplication, use a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}.

Learn more about Radical Operations →

More Radical Operations Examples

Example 1 easy

Simplify [formula].

Example 2 medium

Simplify [formula].

Example 3 medium

Multiply [formula] and simplify.

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