# How do you simplify 5sqrt(-25) • i?

Nov 14, 2015

$- 25$

#### Explanation:

$5 \sqrt{- 25} \cdot i = 5 i \sqrt{25} \cdot i = 5 i \cdot 5 i = 25 \cdot {i}^{2} = - 25$

In common with all non-zero numbers, $- 25$ has two square roots.

The square root represented by the symbols $\sqrt{- 25}$ is the principal square root $i \sqrt{25} = 5 i$. The other square root is $- \sqrt{- 25} = - i \sqrt{25} = - 5 i$.

When $a , b \ge 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$, but that fails if both $a < 0$ and $b < 0$ as in this example:

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