Evaluate square root of -3 square root of -48

$\sqrt{- 3} \sqrt{- 48}$

Step 1

Rewrite $- 3$ as $- 1 (3)$ .

$\sqrt{- 1 (3)} \sqrt{- 48}$

Step 2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$\sqrt{- 1} \cdot \sqrt{3} \sqrt{- 48}$

Step 3

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

$i \cdot \sqrt{3} \sqrt{- 48}$

Step 4

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

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

Step 5

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

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

Step 6

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

$i \cdot \sqrt{3} (i \cdot \sqrt{48})$

Step 7

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

Factor $16$ out of $48$ .

$i \sqrt{3} (i \cdot \sqrt{16 (3)})$

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

$i \sqrt{3} (i \cdot \sqrt{4^{2} \cdot 3})$

Step 8

Pull terms out from under the radical.

$i \sqrt{3} (i \cdot (4 \sqrt{3}))$

Step 9

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

$i \sqrt{3} (4 \cdot i \sqrt{3})$

Step 10

Multiply $i \sqrt{3} (4 i \sqrt{3})$ .

Raise $i$ to the power of $1$ .

$\sqrt{3} (4 (i^{1} i) \sqrt{3})$

Raise $i$ to the power of $1$ .

$\sqrt{3} (4 (i^{1} i^{1}) \sqrt{3})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{3} (4 i^{1 + 1} \sqrt{3})$

Add $1$ and $1$ .

$\sqrt{3} (4 i^{2} \sqrt{3})$

Raise $\sqrt{3}$ to the power of $1$ .

$4 i^{2} ({\sqrt{3}}^{1} \sqrt{3})$

Raise $\sqrt{3}$ to the power of $1$ .

$4 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 i^{2} {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

$4 i^{2} {\sqrt{3}}^{2}$

Step 11

Rewrite $i^{2}$ as $- 1$ .

$4 \cdot - 1 {\sqrt{3}}^{2}$

Step 12

Multiply $4$ by $- 1$ .

$- 4 {\sqrt{3}}^{2}$

Step 13

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- 4 {(3^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 4 \cdot 3^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$- 4 \cdot 3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 4 \cdot 3^{\frac{2}{2}}$

Rewrite the expression.

$- 4 \cdot 3^{1}$

Evaluate the exponent.

$- 4 \cdot 3$

Step 14

Multiply $- 4$ by $3$ .

$- 12$

Evaluate square root of -3 square root of -48

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