The main rules that we use here are :
i) [tex] \sqrt{a \cdot b}= \sqrt{a} \cdot \sqrt{b} [/tex] for nonnegative values a and b.
ii) [tex] \sqrt{a} \cdot \sqrt{a} =a[/tex].
Thus, first 'decompose' the numbers in the radicals into prime factors:
[tex]4 \sqrt{3} \cdot 10 \cdot \sqrt{2\cdot2\cdot3}\cdot \sqrt{2\cdot 3}\cdot \sqrt{2}. [/tex].
By rule (i) we write:
[tex]4 \sqrt{3} \cdot 10 \cdot \sqrt{2}\cdot \sqrt{2}\cdot \sqrt{\cdot3}\cdot \sqrt{2} \cdot \sqrt{3} \cdot \sqrt{2}[/tex].
We can collect these terms as follows:
[tex]40 \sqrt{3}\cdot (\sqrt{3}\cdot \sqrt{3}) \cdot (\sqrt{2}\cdot \sqrt{2}) \cdot( \sqrt{2} \cdot \sqrt{2})[/tex], and by rule (ii) we have:
[tex]40 \sqrt{3}\cdot 3 \cdot 2 \cdot2=40\cdot12\cdot \sqrt{3}=480 \sqrt{3}.[/tex]
Answer: [tex]480 \sqrt{3} [/tex].
