1Cademy - Simplifying $$(5-2\sqrt{3})(5+2\sqrt{3})$$

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

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Simplifying $(5-2\sqrt{3})(5+2\sqrt{3})$

To multiply two binomials that form a conjugate pair involving radicals, such as $(5 - 2\sqrt{3})(5 + 2\sqrt{3})$ , use the Product of Conjugates Pattern: $(a - b)(a + b) = a^2 - b^2$ .

Here, $a = 5$ and $b = 2\sqrt{3}$ . Apply the pattern: $5^2 - (2\sqrt{3})^2$ Square the first term: $5^2 = 25$ . Square the second term by squaring both the numerical coefficient and the radical separately: $(2\sqrt{3})^2 = 2^2 \cdot (\sqrt{3})^2 = 4 \cdot 3 = 12$ . Substitute these values back into the difference of squares: $25 - 12 = 13$ The final product is the integer $13$ , demonstrating that the product of conjugates containing square roots always results in an expression with no radical.

Updated 2026-05-25

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OpenStax Intermediate Algebra

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