# How do you multiply the square root of 3 times the square root of 5?

Oct 13, 2015

If $a , b \ge 0$ then $\sqrt{a} \sqrt{b} = \sqrt{a b}$, so

$\sqrt{3} \sqrt{5} = \sqrt{3 \cdot 5} = \sqrt{15}$

#### Explanation:

${\left(\sqrt{a} \sqrt{b}\right)}^{2} = \sqrt{a} \sqrt{b} \sqrt{a} \sqrt{b} = \sqrt{a} \sqrt{a} \sqrt{b} \sqrt{b} = a b$

So $\sqrt{a} \sqrt{b}$ is always a square root of $a b$, but is it $\sqrt{a b}$ or $- \sqrt{a b}$?

If $a , b \ge 0$ then $\sqrt{a}$, $\sqrt{b}$ and $\sqrt{a b}$ are all Real and non-negative, so $\sqrt{a} \sqrt{b} = \sqrt{a b}$

But if $a , b < 0$ then $\sqrt{a} \sqrt{b} \ne \sqrt{a b}$ as we can see:

$1 = \sqrt{1} = \sqrt{- 1 \cdot - 1} \ne \sqrt{- 1} \cdot \sqrt{- 1} = - 1$