# How do you simplify sqrt(-4) - sqrt(-25)?

Aug 3, 2018

$z = - 3 i \mathmr{and} z = 3 i$

#### Explanation:

We know for the complex numbers : ${i}^{2} = - 1$

Let ,

$z = \sqrt{- 4} - \sqrt{- 25}$

$\implies z = \sqrt{4 \left(- 1\right)} - \sqrt{25 \left(- 1\right)}$

$\implies z = \sqrt{4 {i}^{2}} - \sqrt{25 {i}^{2}}$

$\implies z = \sqrt{{\left(2 i\right)}^{2}} - \sqrt{\left(5 {i}^{2}\right)}$

$\implies z = \left(\pm 2 i\right) - \left(\pm 5 i\right)$

$\implies z = \left(2 i\right) - \left(5 i\right) \mathmr{and} z = \left(- 2 i\right) - \left(- 5 i\right)$

$\implies z = - 3 i \mathmr{and} z = 3 i$

Aug 3, 2018

$\pm 3 i$

#### Explanation:

Recall that $\sqrt{- 1} = i$. With this in mind, we can rewrite this as

$\sqrt{4} \sqrt{- 1} - \left(\sqrt{25} \sqrt{- 1}\right)$

$\implies \pm 2 i - \left(\pm 5 i\right)$

$\implies 2 i - 5 i = - 3 i$ and $- 2 i - \left(- 5 i\right) = 5 i - 2 \equiv 3 i$

Therefore, our solutions are $\pm 3 i$.

Hope this helps!