Simplify square root of 64+ square root of -48+ square root of 25+ square root of -75

$\sqrt{64} + \sqrt{- 48} + \sqrt{25} + \sqrt{- 75}$

Step 1

Simplify each term.

Rewrite $64$ as $8^{2}$ .

$\sqrt{8^{2}} + \sqrt{- 48} + \sqrt{25} + \sqrt{- 75}$

Pull terms out from under the radical, assuming positive real numbers.

$8 + \sqrt{- 48} + \sqrt{25} + \sqrt{- 75}$

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

$8 + \sqrt{- 1 (48)} + \sqrt{25} + \sqrt{- 75}$

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

$8 + \sqrt{- 1} \cdot \sqrt{48} + \sqrt{25} + \sqrt{- 75}$

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

$8 + i \cdot \sqrt{48} + \sqrt{25} + \sqrt{- 75}$

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

Factor $16$ out of $48$ .

$8 + i \cdot \sqrt{16 (3)} + \sqrt{25} + \sqrt{- 75}$

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

$8 + i \cdot \sqrt{4^{2} \cdot 3} + \sqrt{25} + \sqrt{- 75}$

Pull terms out from under the radical.

$8 + i \cdot (4 \sqrt{3}) + \sqrt{25} + \sqrt{- 75}$

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

$8 + 4 \cdot i \sqrt{3} + \sqrt{25} + \sqrt{- 75}$

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

$8 + 4 i \sqrt{3} + \sqrt{5^{2}} + \sqrt{- 75}$

Pull terms out from under the radical, assuming positive real numbers.

$8 + 4 i \sqrt{3} + 5 + \sqrt{- 75}$

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

$8 + 4 i \sqrt{3} + 5 + \sqrt{- 1 (75)}$

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

$8 + 4 i \sqrt{3} + 5 + \sqrt{- 1} \cdot \sqrt{75}$

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

$8 + 4 i \sqrt{3} + 5 + i \cdot \sqrt{75}$

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

Factor $25$ out of $75$ .

$8 + 4 i \sqrt{3} + 5 + i \cdot \sqrt{25 (3)}$

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

$8 + 4 i \sqrt{3} + 5 + i \cdot \sqrt{5^{2} \cdot 3}$

Pull terms out from under the radical.

$8 + 4 i \sqrt{3} + 5 + i \cdot (5 \sqrt{3})$

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

$8 + 4 i \sqrt{3} + 5 + 5 i \sqrt{3}$

Step 2

Simplify by adding terms.

Add $8$ and $5$ .

$4 i \sqrt{3} + 13 + 5 i \sqrt{3}$

Add $4 i \sqrt{3}$ and $5 i \sqrt{3}$ .

$9 i \sqrt{3} + 13$

Reorder $9 i \sqrt{3}$ and $13$ .

$13 + 9 i \sqrt{3}$

Simplify square root of 64+ square root of -48+ square root of 25+ square root of -75

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