Algebra Examples

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

Step 1

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

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

Step 2

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

$\sqrt{- 1} \cdot \sqrt{3} \cdot \sqrt{- 15}$

Step 3

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

$i \cdot \sqrt{3} \cdot \sqrt{- 15}$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Multiply $i \sqrt{3} (i \sqrt{15})$ .

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Step 7.1

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

$i^{1} i \sqrt{3} \sqrt{15}$

Step 7.2

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

$i^{1} i^{1} \sqrt{3} \sqrt{15}$

Step 7.3

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

$i^{1 + 1} \sqrt{3} \sqrt{15}$

Step 7.4

Add $1$ and $1$ .

$i^{2} \sqrt{3} \sqrt{15}$

Step 7.5

Combine using the product rule for radicals.

$i^{2} \sqrt{15 \cdot 3}$

Step 7.6

Multiply $15$ by $3$ .

$i^{2} \sqrt{45}$

Step 8

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

$- 1 \sqrt{45}$

Step 9

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 9.1

Factor $9$ out of $45$ .

$- 1 \sqrt{9 (5)}$

Step 9.2

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

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

Step 10

Pull terms out from under the radical.

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

Step 11

Multiply $3$ by $- 1$ .

$- 3 \sqrt{5}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- 3 \sqrt{5}$

Decimal Form:

$- 6.70820393 \dots$