Simplify the Radical Expression - square root of 25+ square root of -75+ square root of 36- square root of -48

$- \sqrt{25} + \sqrt{- 75} + \sqrt{36} - \sqrt{- 48}$

Step 1

Simplify each term.

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

$- \sqrt{5^{2}} + \sqrt{- 75} + \sqrt{36} - \sqrt{- 48}$

Pull terms out from under the radical.

$- | 5 | + \sqrt{- 75} + \sqrt{36} - \sqrt{- 48}$

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

$- 1 \cdot 5 + \sqrt{- 75} + \sqrt{36} - \sqrt{- 48}$

Multiply $- 1$ by $5$ .

$- 5 + \sqrt{- 75} + \sqrt{36} - \sqrt{- 48}$

Rewrite $- 75$ as $- 1 (75)$ .

$- 5 + \sqrt{- 1 (75)} + \sqrt{36} - \sqrt{- 48}$

Rewrite $\sqrt{- 1 (75)}$ as $\sqrt{- 1} \cdot \sqrt{75}$ .

$- 5 + \sqrt{- 1} \cdot \sqrt{75} + \sqrt{36} - \sqrt{- 48}$

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

$- 5 + i \cdot \sqrt{75} + \sqrt{36} - \sqrt{- 48}$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$- 5 + i \cdot \sqrt{25 (3)} + \sqrt{36} - \sqrt{- 48}$

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

$- 5 + i \cdot \sqrt{5^{2} \cdot 3} + \sqrt{36} - \sqrt{- 48}$

Pull terms out from under the radical.

$- 5 + i \cdot (| 5 | \sqrt{3}) + \sqrt{36} - \sqrt{- 48}$

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

$- 5 + i \cdot (5 \sqrt{3}) + \sqrt{36} - \sqrt{- 48}$

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

$- 5 + 5 \cdot i \sqrt{3} + \sqrt{36} - \sqrt{- 48}$

Rewrite $36$ as $6^{2}$ .

$- 5 + 5 i \sqrt{3} + \sqrt{6^{2}} - \sqrt{- 48}$

Pull terms out from under the radical.

$- 5 + 5 i \sqrt{3} + | 6 | - \sqrt{- 48}$

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$- 5 + 5 i \sqrt{3} + 6 - \sqrt{- 48}$

Rewrite $- 48$ as $- 1 (48)$ .

$- 5 + 5 i \sqrt{3} + 6 - \sqrt{- 1 (48)}$

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$- 5 + 5 i \sqrt{3} + 6 - (\sqrt{- 1} \cdot \sqrt{48})$

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

$- 5 + 5 i \sqrt{3} + 6 - (i \cdot \sqrt{48})$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$- 5 + 5 i \sqrt{3} + 6 - (i \cdot \sqrt{16 (3)})$

Rewrite $16$ as $4^{2}$ .

$- 5 + 5 i \sqrt{3} + 6 - (i \cdot \sqrt{4^{2} \cdot 3})$

Pull terms out from under the radical.

$- 5 + 5 i \sqrt{3} + 6 - (i \cdot (| 4 | \sqrt{3}))$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$- 5 + 5 i \sqrt{3} + 6 - (i \cdot (4 \sqrt{3}))$

Move $4$ to the left of $i$ .

$- 5 + 5 i \sqrt{3} + 6 - (4 \cdot i \sqrt{3})$

Multiply $4$ by $- 1$ .

$- 5 + 5 i \sqrt{3} + 6 - 4 i \sqrt{3}$

Step 2

Add $- 5$ and $6$ .

$5 i \sqrt{3} + 1 - 4 i \sqrt{3}$

Step 3

Subtract $4 i \sqrt{3}$ from $5 i \sqrt{3}$ .

$1 i \sqrt{3} + 1$

Step 4

Multiply $i$ by $1$ .

$i \sqrt{3} + 1$

Step 5

Reorder $i \sqrt{3}$ and $1$ .

$1 + i \sqrt{3}$

Simplify the Radical Expression - square root of 25+ square root of -75+ square root of 36- square root of -48

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