Simplify (3 square root of -5)(-4 square root of -12)

$(3 \sqrt{- 5}) (- 4 \sqrt{- 12})$

Step 1

Rewrite $- 5$ as $- 1 (5)$ .

$3 \sqrt{- 1 (5)} (- 4 \sqrt{- 12})$

Step 2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$3 (\sqrt{- 1} \cdot \sqrt{5}) (- 4 \sqrt{- 12})$

Step 3

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

$3 (i \cdot \sqrt{5}) (- 4 \sqrt{- 12})$

Step 4

Rewrite $- 12$ as $- 1 (12)$ .

$3 (i \cdot \sqrt{5}) (- 4 \sqrt{- 1 (12)})$

Step 5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$3 (i \cdot \sqrt{5}) (- 4 (\sqrt{- 1} \cdot \sqrt{12}))$

Step 6

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

$3 (i \cdot \sqrt{5}) (- 4 (i \cdot \sqrt{12}))$

Step 7

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

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

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

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

Step 8

Pull terms out from under the radical.

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

Step 9

Simplify the expression.

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

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

Multiply $2$ by $- 4$ .

$3 i \sqrt{5} (- 8 (i \sqrt{3}))$

Step 10

Multiply $3 i \sqrt{5} (- 8 i \sqrt{3})$ .

Multiply $- 8$ by $3$ .

$- 24 i \sqrt{5} (i \sqrt{3})$

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

$- 24 (i^{1} i) \sqrt{5} \sqrt{3}$

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

$- 24 (i^{1} i^{1}) \sqrt{5} \sqrt{3}$

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

$- 24 i^{1 + 1} \sqrt{5} \sqrt{3}$

Add $1$ and $1$ .

$- 24 i^{2} \sqrt{5} \sqrt{3}$

Combine using the product rule for radicals.

$- 24 i^{2} \sqrt{3 \cdot 5}$

Multiply $3$ by $5$ .

$- 24 i^{2} \sqrt{15}$

Step 11

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

$- 24 \cdot - 1 \sqrt{15}$

Step 12

Multiply $- 24$ by $- 1$ .

$24 \sqrt{15}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$24 \sqrt{15}$

Decimal Form:

$92.95160030 \dots$

Simplify (3 square root of -5)(-4 square root of -12)

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