Algebra Examples

$\sqrt{- 3} \cdot \sqrt{- 12} \cdot \sqrt{- 4}$

Step 1

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

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

Step 2

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

$\sqrt{- 1} \cdot \sqrt{3} \cdot \sqrt{- 12} \cdot \sqrt{- 4}$

Step 3

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

$i \cdot \sqrt{3} \cdot \sqrt{- 12} \cdot \sqrt{- 4}$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 8

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 9

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

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

Step 10

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

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Step 10.1

Factor $4$ out of $12$ .

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

Step 10.2

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

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

Step 11

Pull terms out from under the radical.

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

Step 12

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

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

Step 13

Multiply $i \sqrt{3} (2 i \sqrt{3})$ .

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

Add $1$ and $1$ .

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

Step 13.5

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

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

Step 13.6

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

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

Step 13.7

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

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

Step 13.8

Add $1$ and $1$ .

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

Step 14

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

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

Step 15

Multiply $2$ by $- 1$ .

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

Step 16

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

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Step 16.1

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

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

Step 16.2

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

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

Step 16.3

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

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

Step 16.4

Cancel the common factor of $2$ .

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Step 16.4.1

Cancel the common factor.

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

Step 16.4.2

Rewrite the expression.

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

Step 16.5

Evaluate the exponent.

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

Step 17

Write the expression using exponents.

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Step 17.1

Multiply $- 2$ by $3$ .

$- 6 \cdot (i \cdot \sqrt{4})$

Step 17.2

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

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

Step 18

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

$- 6 \cdot (i \cdot 2)$

Step 19

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

$- 6 \cdot (2 \cdot i)$

Step 20

Multiply $2$ by $- 6$ .

$- 12 i$