Simplify the Radical Expression - square root of 25+ square root of -24+ square root of 1+ square root of -54

$- \sqrt{25} + \sqrt{- 24} + \sqrt{1} + \sqrt{- 54}$

Step 1

Simplify each term.

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

$- \sqrt{5^{2}} + \sqrt{- 24} + \sqrt{1} + \sqrt{- 54}$

Pull terms out from under the radical.

$- | 5 | + \sqrt{- 24} + \sqrt{1} + \sqrt{- 54}$

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

$- 1 \cdot 5 + \sqrt{- 24} + \sqrt{1} + \sqrt{- 54}$

Multiply $- 1$ by $5$ .

$- 5 + \sqrt{- 24} + \sqrt{1} + \sqrt{- 54}$

Rewrite $- 24$ as $- 1 (24)$ .

$- 5 + \sqrt{- 1 (24)} + \sqrt{1} + \sqrt{- 54}$

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$- 5 + \sqrt{- 1} \cdot \sqrt{24} + \sqrt{1} + \sqrt{- 54}$

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

$- 5 + i \cdot \sqrt{24} + \sqrt{1} + \sqrt{- 54}$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$- 5 + i \cdot \sqrt{4 (6)} + \sqrt{1} + \sqrt{- 54}$

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

$- 5 + i \cdot \sqrt{2^{2} \cdot 6} + \sqrt{1} + \sqrt{- 54}$

Pull terms out from under the radical.

$- 5 + i \cdot (| 2 | \sqrt{6}) + \sqrt{1} + \sqrt{- 54}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$- 5 + i \cdot (2 \sqrt{6}) + \sqrt{1} + \sqrt{- 54}$

Move $2$ to the left of $i$ .

$- 5 + 2 \cdot i \sqrt{6} + \sqrt{1} + \sqrt{- 54}$

Any root of $1$ is $1$ .

$- 5 + 2 i \sqrt{6} + 1 + \sqrt{- 54}$

Rewrite $- 54$ as $- 1 (54)$ .

$- 5 + 2 i \sqrt{6} + 1 + \sqrt{- 1 (54)}$

Rewrite $\sqrt{- 1 (54)}$ as $\sqrt{- 1} \cdot \sqrt{54}$ .

$- 5 + 2 i \sqrt{6} + 1 + \sqrt{- 1} \cdot \sqrt{54}$

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

$- 5 + 2 i \sqrt{6} + 1 + i \cdot \sqrt{54}$

Rewrite $54$ as $3^{2} \cdot 6$ .

Factor $9$ out of $54$ .

$- 5 + 2 i \sqrt{6} + 1 + i \cdot \sqrt{9 (6)}$

Rewrite $9$ as $3^{2}$ .

$- 5 + 2 i \sqrt{6} + 1 + i \cdot \sqrt{3^{2} \cdot 6}$

Pull terms out from under the radical.

$- 5 + 2 i \sqrt{6} + 1 + i \cdot (| 3 | \sqrt{6})$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$- 5 + 2 i \sqrt{6} + 1 + i \cdot (3 \sqrt{6})$

Move $3$ to the left of $i$ .

$- 5 + 2 i \sqrt{6} + 1 + 3 i \sqrt{6}$

Step 2

Add $- 5$ and $1$ .

$2 i \sqrt{6} - 4 + 3 i \sqrt{6}$

Step 3

Add $2 i \sqrt{6}$ and $3 i \sqrt{6}$ .

$5 i \sqrt{6} - 4$

Step 4

Reorder $5 i \sqrt{6}$ and $- 4$ .

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

Simplify the Radical Expression - square root of 25+ square root of -24+ square root of 1+ square root of -54

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