1Cademy - Simplifying $$(4 - \sqrt{2})(4 + \sqrt{2})$$

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Product of Conjugates Pattern

Example

Simplifying $(4 - \sqrt{2})(4 + \sqrt{2})$

Apply the Product of Conjugates Pattern to multiply two binomials that are conjugates containing a square root.

Recognize the expression as a product of conjugates: both binomials share the same first term $4$ and the same last term $\sqrt{2}$ , with one using subtraction and the other addition. Use the formula $(a - b)(a + b) = a^2 - b^2$ , where $a = 4$ and $b = \sqrt{2}$ .

Step 1 — Square the first term: $4^2 = 16$ .

Step 2 — Square the last term: $(\sqrt{2})^2 = 2$ .

Step 3 — Write the difference of squares:

$(4 - \sqrt{2})(4 + \sqrt{2}) = 16 - 2 = 14$

The result is the integer 14. When the Product of Conjugates Pattern is applied to a conjugate pair containing a square root, the radical is eliminated entirely — squaring the square root returns the radicand, and the final answer is a rational number with no remaining radical. This property of conjugates is especially useful for rationalizing denominators, since multiplying a binomial containing a square root by its conjugate always produces a rational result.

Updated 2026-05-01

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