Question

multiply.
$(9\sqrt{2} + 4\sqrt{6})(3\sqrt{6} + 5\sqrt{2})$
simplify your answer as much as possible.

Explanation:

Step1: Apply the distributive property (FOIL method)

$$(9\sqrt{2}+4\sqrt{6})(3\sqrt{6}+5\sqrt{2}) = 9\sqrt{2}\times3\sqrt{6} + 9\sqrt{2}\times5\sqrt{2} + 4\sqrt{6}\times3\sqrt{6} + 4\sqrt{6}\times5\sqrt{2}$$

Step2: Simplify each term

• For \(9\sqrt{2}\times3\sqrt{6}\): \(9\times3\times\sqrt{2\times6}=27\sqrt{12}=27\times2\sqrt{3}=54\sqrt{3}\)
• For \(9\sqrt{2}\times5\sqrt{2}\): \(9\times5\times\sqrt{2\times2}=45\times2 = 90\)
• For \(4\sqrt{6}\times3\sqrt{6}\): \(4\times3\times\sqrt{6\times6}=12\times6 = 72\)
• For \(4\sqrt{6}\times5\sqrt{2}\): \(4\times5\times\sqrt{6\times2}=20\sqrt{12}=20\times2\sqrt{3}=40\sqrt{3}\)

Step3: Combine like terms

• Combine the \(\sqrt{3}\) terms: \(54\sqrt{3}+40\sqrt{3}=94\sqrt{3}\)
• Combine the constant terms: \(90 + 72=162\)

Step4: Combine all terms

$$94\sqrt{3}+162$$

Answer:

\(162 + 94\sqrt{3}\)