Question

Multiply.

Answer 1

The main rules that we use here are :

i) $\sqrt{a \cdot b}= \sqrt{a} \cdot \sqrt{b}$ for nonnegative values a and b.

ii) $\sqrt{a} \cdot \sqrt{a} =a$.


Thus, first 'decompose' the numbers in the radicals into prime factors:

$4 \sqrt{3} \cdot 10 \cdot \sqrt{2\cdot2\cdot3}\cdot \sqrt{2\cdot 3}\cdot \sqrt{2}.$.

By rule (i) we write:

$4 \sqrt{3} \cdot 10 \cdot \sqrt{2}\cdot \sqrt{2}\cdot \sqrt{\cdot3}\cdot \sqrt{2} \cdot \sqrt{3} \cdot \sqrt{2}$.

We can collect these terms as follows:

$40 \sqrt{3}\cdot (\sqrt{3}\cdot \sqrt{3}) \cdot (\sqrt{2}\cdot \sqrt{2}) \cdot( \sqrt{2} \cdot \sqrt{2})$, and by rule (ii) we have:

$40 \sqrt{3}\cdot 3 \cdot 2 \cdot2=40\cdot12\cdot \sqrt{3}=480 \sqrt{3}.$


Answer: $480 \sqrt{3}$.