Algebra Examples

$\sqrt{- 6 x} \cdot \sqrt{21 x^{3}}$

Step 1

Rewrite $21 x^{3}$ as $x^{2} \cdot (21 x)$ .

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Step 1.1

Factor out $x^{2}$ .

$\sqrt{- 6 x} \cdot \sqrt{21 (x^{2} x)}$

Step 1.2

Reorder $21$ and $x^{2}$ .

$\sqrt{- 6 x} \cdot \sqrt{x^{2} \cdot 21 x}$

Step 1.3

Add parentheses.

$\sqrt{- 6 x} \cdot \sqrt{x^{2} \cdot (21 x)}$

Step 2

Pull terms out from under the radical.

$\sqrt{- 6 x} \cdot (x \sqrt{21 x})$

Step 3

Multiply $\sqrt{- 6 x} (x \sqrt{21 x})$ .

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Step 3.1

Combine using the product rule for radicals.

$x \sqrt{- 6 x (21 x)}$

Step 3.2

Multiply $21$ by $- 6$ .

$x \sqrt{- 126 x \cdot x}$

Step 3.3

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

$x \sqrt{- 126 (x^{1} x)}$

Step 3.4

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

$x \sqrt{- 126 (x^{1} x^{1})}$

Step 3.5

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

$x \sqrt{- 126 x^{1 + 1}}$

Step 3.6

Add $1$ and $1$ .

$x \sqrt{- 126 x^{2}}$

Step 4

Rewrite $- 126 x^{2}$ as ${(3 x)}^{2} \cdot - 14$ .

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Step 4.1

Factor $9$ out of $- 126$ .

$x \sqrt{9 (- 14) x^{2}}$

Step 4.2

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

$x \sqrt{3^{2} \cdot - 14 x^{2}}$

Step 4.3

Move $- 14$ .

$x \sqrt{3^{2} x^{2} \cdot - 14}$

Step 4.4

Rewrite $3^{2} x^{2}$ as ${(3 x)}^{2}$ .

$x \sqrt{{(3 x)}^{2} \cdot - 14}$

Step 5

Pull terms out from under the radical.

$x (3 x \sqrt{- 14})$

Step 6

Rewrite using the commutative property of multiplication.

$3 \sqrt{- 14} x \cdot x$

Step 7

Rewrite $- 14$ as $- 1 (14)$ .

$3 \sqrt{- 1 (14)} x \cdot x$

Step 8

Rewrite $\sqrt{- 1 (14)}$ as $\sqrt{- 1} \cdot \sqrt{14}$ .

$3 (\sqrt{- 1} \cdot \sqrt{14}) x \cdot x$

Step 9

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

$3 (i \cdot \sqrt{14}) x \cdot x$

Step 10

Multiply $x$ by $x$ by adding the exponents.

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Step 10.1

Move $x$ .

$3 (i \cdot \sqrt{14}) (x \cdot x)$

Step 10.2

Multiply $x$ by $x$ .

$3 (i \cdot \sqrt{14}) x^{2}$

$3 i \sqrt{14} x^{2}$