Question

multiply.
$(4\sqrt{2} + 3)(3\sqrt{10} - 2)$

Explanation:

Step1: Apply the distributive property (FOIL method)

We need to multiply each term in the first binomial \((4\sqrt{2} + 3)\) by each term in the second binomial \((3\sqrt{10}-2)\).
\[

$$\begin{align*} &(4\sqrt{2}+3)(3\sqrt{10}-2)\\ =&4\sqrt{2}\times3\sqrt{10}+4\sqrt{2}\times(-2)+3\times3\sqrt{10}+3\times(-2) \end{align*}$$

\]

Step2: Simplify each term

• For the first term \(4\sqrt{2}\times3\sqrt{10}\):

Using the property \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\) and \(a\times b\times\sqrt{c}\times\sqrt{d}=ab\sqrt{cd}\), we have \(4\times3\times\sqrt{2\times10} = 12\sqrt{20}\). And \(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\), so \(12\sqrt{20}=12\times2\sqrt{5} = 24\sqrt{5}\).

• For the second term \(4\sqrt{2}\times(-2)=-8\sqrt{2}\)
• For the third term \(3\times3\sqrt{10}=9\sqrt{10}\)
• For the fourth term \(3\times(-2)=-6\)

Putting it all together:
\[24\sqrt{5}-8\sqrt{2}+9\sqrt{10}-6\]

Answer:

\(24\sqrt{5}-8\sqrt{2}+9\sqrt{10}-6\)