Algebra Examples

$3 \sqrt{- 24} \cdot 2 \sqrt{- 15}$

Step 1

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

$3 \sqrt{- 1 (24)} \cdot (2 \sqrt{- 15})$

Step 2

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

$3 (\sqrt{- 1} \cdot \sqrt{24}) \cdot (2 \sqrt{- 15})$

Step 3

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

$3 (i \cdot \sqrt{24}) \cdot (2 \sqrt{- 15})$

Step 4

Rewrite $- 15$ as $- 1 (15)$ .

$3 (i \cdot \sqrt{24}) \cdot (2 \sqrt{- 1 (15)})$

Step 5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$3 (i \cdot \sqrt{24}) \cdot (2 (\sqrt{- 1} \cdot \sqrt{15}))$

Step 6

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

$3 (i \cdot \sqrt{24}) \cdot (2 (i \cdot \sqrt{15}))$

Step 7

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

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Step 7.1

Factor $4$ out of $24$ .

$3 (i \cdot \sqrt{4 (6)}) \cdot (2 (i \cdot \sqrt{15}))$

Step 7.2

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

$3 (i \cdot \sqrt{2^{2} \cdot 6}) \cdot (2 (i \cdot \sqrt{15}))$

Step 8

Pull terms out from under the radical.

$3 (i \cdot (2 \sqrt{6})) \cdot (2 (i \cdot \sqrt{15}))$

Step 9

Simplify the expression.

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Step 9.1

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

$3 (2 \cdot i \sqrt{6}) \cdot (2 (i \cdot \sqrt{15}))$

Step 9.2

Multiply $2$ by $3$ .

$6 (i \sqrt{6}) \cdot (2 (i \cdot \sqrt{15}))$

Step 10

Multiply $6 i \sqrt{6} (2 i \sqrt{15})$ .

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Step 10.1

Multiply $2$ by $6$ .

$12 i \sqrt{6} (i \sqrt{15})$

Step 10.2

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

$12 (i^{1} i) \sqrt{6} \sqrt{15}$

Step 10.3

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

$12 (i^{1} i^{1}) \sqrt{6} \sqrt{15}$

Step 10.4

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

$12 i^{1 + 1} \sqrt{6} \sqrt{15}$

Step 10.5

Add $1$ and $1$ .

$12 i^{2} \sqrt{6} \sqrt{15}$

Step 10.6

Combine using the product rule for radicals.

$12 i^{2} \sqrt{15 \cdot 6}$

Step 10.7

Multiply $15$ by $6$ .

$12 i^{2} \sqrt{90}$

Step 11

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

$12 \cdot - 1 \sqrt{90}$

Step 12

Multiply $12$ by $- 1$ .

$- 12 \sqrt{90}$

Step 13

Rewrite $90$ as $3^{2} \cdot 10$ .

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Step 13.1

Factor $9$ out of $90$ .

$- 12 \sqrt{9 (10)}$

Step 13.2

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

$- 12 \sqrt{3^{2} \cdot 10}$

Step 14

Pull terms out from under the radical.

$- 12 (3 \sqrt{10})$

Step 15

Multiply $3$ by $- 12$ .

$- 36 \sqrt{10}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$- 36 \sqrt{10}$

Decimal Form:

$- 113.84199576 \dots$