第1465题：complex number

When multiplying and dividing complex numbers we must take care to understand that the product and quotient rules for radicals require that bath $a$ and $b$ are positive. In other words, if $\sqrt[n]{a}$ and $\sqrt[n]{b}$  are both real numbers then we have the following rules.

Product rule for radicals:

$\sqrt[n]{a \cdot b}=\sqrt[n]{a} \cdot \sqrt[n]{b}$

Quotient rule for radicals:

$\sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$

However, when $a$ and $b$ are both negative the product rule and quotient rule fo radicals fails to get a true statement.

For example, Multiply:

$\sqrt{-5} \cdot \sqrt{-12}$

Which of the following two solutions is correct?

A.

$\sqrt{-5} \cdot \sqrt{-12}$

$=\sqrt{-5*-12}$

$=\sqrt{60}$

$=2\sqrt{15}$

B.

$\sqrt{-5} \cdot \sqrt{-12}$

$=\mathrm{i} \sqrt{5} \cdot \mathrm{i} \sqrt{12}$

$=\mathrm{i}^2\sqrt{60}$