Question

multiply.
\\((7\sqrt{2} - \sqrt{6})(3\sqrt{6} - 2\sqrt{2})\\)
simplify your answer as much as possible.

Explanation:

Step1: Apply distributive property (FOIL)

Multiply each term in the first binomial by each term in the second binomial:
$$(7\sqrt{2})(3\sqrt{6}) + (7\sqrt{2})(-2\sqrt{2}) + (-\sqrt{6})(3\sqrt{6}) + (-\sqrt{6})(-2\sqrt{2})$$

Step2: Simplify each product

• For \( (7\sqrt{2})(3\sqrt{6}) \): Multiply coefficients and radicals: \( 7\times3\times\sqrt{2\times6} = 21\sqrt{12} = 21\times2\sqrt{3} = 42\sqrt{3} \) (since \( \sqrt{12} = 2\sqrt{3} \))
• For \( (7\sqrt{2})(-2\sqrt{2}) \): Multiply coefficients and radicals: \( 7\times(-2)\times\sqrt{2\times2} = -14\times2 = -28 \) (since \( \sqrt{2\times2} = 2 \))
• For \( (-\sqrt{6})(3\sqrt{6}) \): Multiply coefficients and radicals: \( -1\times3\times\sqrt{6\times6} = -3\times6 = -18 \) (since \( \sqrt{6\times6} = 6 \))
• For \( (-\sqrt{6})(-2\sqrt{2}) \): Multiply coefficients and radicals: \( (-1)\times(-2)\times\sqrt{6\times2} = 2\sqrt{12} = 2\times2\sqrt{3} = 4\sqrt{3} \) (since \( \sqrt{12} = 2\sqrt{3} \))

Step3: Combine like terms

Combine the constant terms and the radical terms:

• Constant terms: \( -28 - 18 = -46 \)
• Radical terms: \( 42\sqrt{3} + 4\sqrt{3} = 46\sqrt{3} \)

Step4: Combine all terms

Add the constant and radical terms together:
\( 46\sqrt{3} - 46 \)

Answer:

\( 46\sqrt{3} - 46 \)