Algebra Examples

$\sqrt[3]{- 6 x^{5}} \cdot \sqrt[3]{- 18 x^{4}}$

Step 1

Rewrite $- 6 x^{5}$ as ${(- x)}^{3} \cdot (6 x^{2})$ .

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Step 1.1

Rewrite $- 6$ as $- 1 (6)$ .

$\sqrt[3]{- 1 (6) x^{5}} \cdot \sqrt[3]{- 18 x^{4}}$

Step 1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \cdot 6 x^{5}} \cdot \sqrt[3]{- 18 x^{4}}$

Step 1.3

Factor out $x^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \cdot 6 (x^{3} x^{2})} \cdot \sqrt[3]{- 18 x^{4}}$

Step 1.4

Move $6$ .

$\sqrt[3]{{(- 1)}^{3} x^{3} \cdot 6 x^{2}} \cdot \sqrt[3]{- 18 x^{4}}$

Step 1.5

Rewrite ${(- 1)}^{3} x^{3}$ as ${(- x)}^{3}$ .

$\sqrt[3]{{(- x)}^{3} \cdot 6 x^{2}} \cdot \sqrt[3]{- 18 x^{4}}$

Step 1.6

Add parentheses.

$\sqrt[3]{{(- x)}^{3} \cdot (6 x^{2})} \cdot \sqrt[3]{- 18 x^{4}}$

Step 2

Pull terms out from under the radical.

$- x \sqrt[3]{6 x^{2}} \cdot \sqrt[3]{- 18 x^{4}}$

Step 3

Rewrite $- 18 x^{4}$ as ${(- x)}^{3} \cdot (18 x)$ .

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Step 3.1

Rewrite $- 18$ as $- 1 (18)$ .

$- x \sqrt[3]{6 x^{2}} \cdot \sqrt[3]{- 1 (18) x^{4}}$

Step 3.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$- x \sqrt[3]{6 x^{2}} \cdot \sqrt[3]{{(- 1)}^{3} \cdot 18 x^{4}}$

Step 3.3

Factor out $x^{3}$ .

$- x \sqrt[3]{6 x^{2}} \cdot \sqrt[3]{{(- 1)}^{3} \cdot 18 (x^{3} x)}$

Step 3.4

Move $18$ .

$- x \sqrt[3]{6 x^{2}} \cdot \sqrt[3]{{(- 1)}^{3} x^{3} \cdot 18 x}$

Step 3.5

Rewrite ${(- 1)}^{3} x^{3}$ as ${(- x)}^{3}$ .

$- x \sqrt[3]{6 x^{2}} \cdot \sqrt[3]{{(- x)}^{3} \cdot 18 x}$

Step 3.6

Add parentheses.

$- x \sqrt[3]{6 x^{2}} \cdot \sqrt[3]{{(- x)}^{3} \cdot (18 x)}$

Step 4

Pull terms out from under the radical.

$- x \sqrt[3]{6 x^{2}} \cdot (- x \sqrt[3]{18 x})$

Step 5

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

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Step 5.1

Move $x$ .

$- (x \cdot x) \sqrt[3]{6 x^{2}} \cdot (- 1 \sqrt[3]{18 x})$

Step 5.2

Multiply $x$ by $x$ .

$- x^{2} \sqrt[3]{6 x^{2}} \cdot (- 1 \sqrt[3]{18 x})$

Step 6

Rewrite $- 1 \sqrt[3]{18 x}$ as $- \sqrt[3]{18 x}$ .

$- x^{2} \sqrt[3]{6 x^{2}} \cdot (- \sqrt[3]{18 x})$

Step 7

Multiply $- x^{2} \sqrt[3]{6 x^{2}} (- \sqrt[3]{18 x})$ .

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Step 7.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} \sqrt[3]{6 x^{2}} \sqrt[3]{18 x}$

Step 7.2

Multiply $x^{2}$ by $1$ .

$x^{2} \sqrt[3]{6 x^{2}} \sqrt[3]{18 x}$

Step 7.3

Combine using the product rule for radicals.

$x^{2} \sqrt[3]{18 x (6 x^{2})}$

Step 7.4

Multiply $6$ by $18$ .

$x^{2} \sqrt[3]{108 x \cdot x^{2}}$

Step 7.5

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 7.5.1

Move $x^{2}$ .

$x^{2} \sqrt[3]{108 (x^{2} x)}$

Step 7.5.2

Multiply $x^{2}$ by $x$ .

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Step 7.5.2.1

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

$x^{2} \sqrt[3]{108 (x^{2} x^{1})}$

Step 7.5.2.2

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

$x^{2} \sqrt[3]{108 x^{2 + 1}}$

Step 7.5.3

Add $2$ and $1$ .

$x^{2} \sqrt[3]{108 x^{3}}$

Step 8

Rewrite $108 x^{3}$ as ${(3 x)}^{3} \cdot 4$ .

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Step 8.1

Factor $27$ out of $108$ .

$x^{2} \sqrt[3]{27 (4) x^{3}}$

Step 8.2

Rewrite $27$ as $3^{3}$ .

$x^{2} \sqrt[3]{3^{3} \cdot 4 x^{3}}$

Step 8.3

Move $4$ .

$x^{2} \sqrt[3]{3^{3} x^{3} \cdot 4}$

Step 8.4

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

$x^{2} \sqrt[3]{{(3 x)}^{3} \cdot 4}$

Step 9

Pull terms out from under the radical.

$x^{2} (3 x \sqrt[3]{4})$

Step 10

Rewrite using the commutative property of multiplication.

$3 \sqrt[3]{4} x^{2} x$

Step 11

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 11.1

Move $x$ .

$3 \sqrt[3]{4} (x \cdot x^{2})$

Step 11.2

Multiply $x$ by $x^{2}$ .

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Step 11.2.1

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

$3 \sqrt[3]{4} (x^{1} x^{2})$

Step 11.2.2

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

$3 \sqrt[3]{4} x^{1 + 2}$

Step 11.3

Add $1$ and $2$ .

$3 \sqrt[3]{4} x^{3}$