Question

recall that the product \\((a + b)(a - b)\\) is the difference of squares, \\(a^2 - b^2\\).
examine this same product for radical expressions.

choose the product of \\((3 + \sqrt{7})(3 - \sqrt{7})\\)

\\(6 - 3\sqrt{7} + 3\sqrt{7} - \sqrt{14}\\)
\\(9 - \sqrt{21} + \sqrt{21} - \sqrt{49}\\)
\\(9 - 3\sqrt{7} + 3\sqrt{7} - \sqrt{49}\\)

Explanation:

Identify the terms for expansion

Using the Distributive Property

$$ (a + b)(c + d) = ac + ad + bc + bd $$

For the expression \((3 + \sqrt{7})(3 - \sqrt{7})\), we set:

$$ a = 3,\quad b = \sqrt{7},\quad c = 3,\quad d = -\sqrt{7} $$

Expand the product term-by-term

Using the Distributive Property

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Simplify the radical product term

Using the Product Property of Radicals

$$ \sqrt{7} \cdot \sqrt{7} = \sqrt{7 \cdot 7} = \sqrt{49} $$

Substituting this back into the expanded expression yields:

$$ 9 - 3\sqrt{7} + 3\sqrt{7} - \sqrt{49} $$

Answer:

• (A) \(6 - 3\sqrt{7} + 3\sqrt{7} - \sqrt{14}\)
• (B) \(9 - \sqrt{21} + \sqrt{21} - \sqrt{49}\)
• (C) \(9 - 3\sqrt{7} + 3\sqrt{7} - \sqrt{49}\) (Correct answer)