1Cademy - Simplifying $$(10+\sqrt{2})^2$$ and $$(1+3\sqrt{6})^2$$

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Binomial Squares Pattern

Example

Simplifying $(10+\sqrt{2})^2$ and $(1+3\sqrt{6})^2$

Apply the Binomial Squares Pattern $(a + b)^2 = a^2 + 2ab + b^2$ to expand each squared binomial containing a square root.

For $(10 + \sqrt{2})^2$ , set $a = 10$ and $b = \sqrt{2}$ : $10^2 + 2(10)(\sqrt{2}) + (\sqrt{2})^2 = 100 + 20\sqrt{2} + 2$ Combine the integers to obtain the simplified expression $102 + 20\sqrt{2}$ .

For $(1 + 3\sqrt{6})^2$ , set $a = 1$ and $b = 3\sqrt{6}$ : $1^2 + 2(1)(3\sqrt{6}) + (3\sqrt{6})^2$ When squaring the last term, square both the coefficient and the radicand: $3^2 \cdot (\sqrt{6})^2 = 9 \cdot 6 = 54$ . The expansion simplifies to: $1 + 6\sqrt{6} + 54$ Combining the integers yields $55 + 6\sqrt{6}$ .