Algebra Examples

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

Step 1

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

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

Step 2

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

$\sqrt{- 1} \cdot \sqrt{24} \cdot \sqrt{- 3} \cdot \sqrt{- 2}$

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 8

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

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

Step 9

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

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

Step 10

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

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Step 10.1

Factor $4$ out of $24$ .

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

Step 10.2

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

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

Step 11

Pull terms out from under the radical.

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

Step 12

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

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

Step 13

Multiply $2 i \sqrt{6} (i \sqrt{3})$ .

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Step 13.1

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

$2 (i^{1} i) \sqrt{6} \sqrt{3} \cdot (i \cdot \sqrt{2})$

Step 13.2

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

$2 (i^{1} i^{1}) \sqrt{6} \sqrt{3} \cdot (i \cdot \sqrt{2})$

Step 13.3

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

$2 i^{1 + 1} \sqrt{6} \sqrt{3} \cdot (i \cdot \sqrt{2})$

Step 13.4

Add $1$ and $1$ .

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

Step 13.5

Combine using the product rule for radicals.

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

Step 13.6

Multiply $3$ by $6$ .

$2 i^{2} \sqrt{18} \cdot (i \cdot \sqrt{2})$

Step 14

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

$2 \cdot - 1 \sqrt{18} \cdot (i \cdot \sqrt{2})$

Step 15

Multiply $2$ by $- 1$ .

$- 2 \sqrt{18} \cdot (i \cdot \sqrt{2})$

Step 16

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 16.1

Factor $9$ out of $18$ .

$- 2 \sqrt{9 (2)} \cdot (i \cdot \sqrt{2})$

Step 16.2

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

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

Step 17

Pull terms out from under the radical.

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

Step 18

Multiply $3$ by $- 2$ .

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

Step 19

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

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Step 19.1

Raise $\sqrt{2}$ to the power of $1$ .

$- 6 ({\sqrt{2}}^{1} \sqrt{2}) i$

Step 19.2

Raise $\sqrt{2}$ to the power of $1$ .

$- 6 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) i$

Step 19.3

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

$- 6 {\sqrt{2}}^{1 + 1} i$

Step 19.4

Add $1$ and $1$ .

$- 6 {\sqrt{2}}^{2} i$

Step 20

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

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Step 20.1

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

$- 6 {(2^{\frac{1}{2}})}^{2} i$

Step 20.2

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

$- 6 \cdot 2^{\frac{1}{2} \cdot 2} i$

Step 20.3

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

$- 6 \cdot 2^{\frac{2}{2}} i$

Step 20.4

Cancel the common factor of $2$ .

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Step 20.4.1

Cancel the common factor.

$- 6 \cdot 2^{\frac{2}{2}} i$

Step 20.4.2

Rewrite the expression.

$- 6 \cdot 2^{1} i$

Step 20.5

Evaluate the exponent.

$- 6 \cdot 2 i$

Step 21

Multiply $- 6$ by $2$ .

$- 12 i$