Simplify the Radical Expression square root of 100+ square root of -1+ square root of 1- square root of -25

$\sqrt{100} + \sqrt{- 1} + \sqrt{1} - \sqrt{- 25}$

Step 1

Simplify each term.

Rewrite $100$ as $10^{2}$ .

$\sqrt{10^{2}} + \sqrt{- 1} + \sqrt{1} - \sqrt{- 25}$

Pull terms out from under the radical.

$| 10 | + \sqrt{- 1} + \sqrt{1} - \sqrt{- 25}$

The absolute value is the distance between a number and zero. The distance between $0$ and $10$ is $10$ .

$10 + \sqrt{- 1} + \sqrt{1} - \sqrt{- 25}$

Rewrite $\sqrt{- 1}$ as $i$ .

$10 + i + \sqrt{1} - \sqrt{- 25}$

Any root of $1$ is $1$ .

$10 + i + 1 - \sqrt{- 25}$

Rewrite $- 25$ as $- 1 (25)$ .

$10 + i + 1 - \sqrt{- 1 (25)}$

Rewrite $\sqrt{- 1 (25)}$ as $\sqrt{- 1} \cdot \sqrt{25}$ .

$10 + i + 1 - (\sqrt{- 1} \cdot \sqrt{25})$

Rewrite $\sqrt{- 1}$ as $i$ .

$10 + i + 1 - (i \cdot \sqrt{25})$

Rewrite $25$ as $5^{2}$ .

$10 + i + 1 - (i \cdot \sqrt{5^{2}})$

Pull terms out from under the radical.

$10 + i + 1 - (i \cdot | 5 |)$

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$10 + i + 1 - (i \cdot 5)$

Move $5$ to the left of $i$ .

$10 + i + 1 - (5 \cdot i)$

Multiply $5$ by $- 1$ .

$10 + i + 1 - 5 i$

Step 2

Add $10$ and $1$ .

$11 + i - 5 i$

Step 3

Subtract $5 i$ from $i$ .

$11 - 4 i$