Algebra Examples

$\sqrt[3]{- 3 x^{2}} \cdot \sqrt[3]{24 x^{5}}$

Step 1

Rewrite $- 3 x^{2}$ as ${({(- 1)}^{3})}^{3} \cdot (3 x^{2})$ .

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Step 1.1

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

$\sqrt[3]{- 1 (3) x^{2}} \cdot \sqrt[3]{24 x^{5}}$

Step 1.2

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

$\sqrt[3]{{(- 1)}^{3} \cdot 3 x^{2}} \cdot \sqrt[3]{24 x^{5}}$

Step 1.3

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

$\sqrt[3]{{({(- 1)}^{3})}^{3} \cdot 3 x^{2}} \cdot \sqrt[3]{24 x^{5}}$

Step 1.4

Add parentheses.

$\sqrt[3]{{({(- 1)}^{3})}^{3} \cdot (3 x^{2})} \cdot \sqrt[3]{24 x^{5}}$

Step 2

Pull terms out from under the radical.

${(- 1)}^{3} \sqrt[3]{3 x^{2}} \cdot \sqrt[3]{24 x^{5}}$

Step 3

Simplify the expression.

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Step 3.1

Raise $- 1$ to the power of $3$ .

$- 1 \sqrt[3]{3 x^{2}} \cdot \sqrt[3]{24 x^{5}}$

Step 3.2

Rewrite $- 1 \sqrt[3]{3 x^{2}}$ as $- \sqrt[3]{3 x^{2}}$ .

$- \sqrt[3]{3 x^{2}} \cdot \sqrt[3]{24 x^{5}}$

Step 4

Rewrite $24 x^{5}$ as ${(2 x)}^{3} \cdot (3 x^{2})$ .

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Step 4.1

Factor $8$ out of $24$ .

$- \sqrt[3]{3 x^{2}} \cdot \sqrt[3]{8 (3) x^{5}}$

Step 4.2

Rewrite $8$ as $2^{3}$ .

$- \sqrt[3]{3 x^{2}} \cdot \sqrt[3]{2^{3} \cdot 3 x^{5}}$

Step 4.3

Factor out $x^{3}$ .

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

Step 4.4

Move $3$ .

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

Step 4.5

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

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

Step 4.6

Add parentheses.

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

Step 5

Pull terms out from under the radical.

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

Step 6

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

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Step 6.1

Multiply $2$ by $- 1$ .

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

Step 6.2

Raise $\sqrt[3]{3 x^{2}}$ to the power of $1$ .

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

Step 6.3

Raise $\sqrt[3]{3 x^{2}}$ to the power of $1$ .

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 7

Rewrite ${\sqrt[3]{3 x^{2}}}^{2}$ as $\sqrt[3]{{(3 x^{2})}^{2}}$ .

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

Step 8

Apply the product rule to $3 x^{2}$ .

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

Step 9

Raise $3$ to the power of $2$ .

$- 2 (x \sqrt[3]{9 {(x^{2})}^{2}})$

Step 10

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 10.1

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

$- 2 (x \sqrt[3]{9 x^{2 \cdot 2}})$

Step 10.2

Multiply $2$ by $2$ .

$- 2 (x \sqrt[3]{9 x^{4}})$

Step 11

Rewrite $9 x^{4}$ as $x^{3} \cdot (9 x)$ .

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Step 11.1

Factor out $x^{3}$ .

$- 2 (x \sqrt[3]{9 (x^{3} x)})$

Step 11.2

Reorder $9$ and $x^{3}$ .

$- 2 (x \sqrt[3]{x^{3} \cdot 9 x})$

Step 11.3

Add parentheses.

$- 2 (x \sqrt[3]{x^{3} \cdot (9 x)})$

Step 12

Pull terms out from under the radical.

$- 2 (x (x \sqrt[3]{9 x}))$

Step 13

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

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Step 13.1

Move $x$ .

$- 2 (x \cdot x \sqrt[3]{9 x})$

Step 13.2

Multiply $x$ by $x$ .

$- 2 (x^{2} \sqrt[3]{9 x})$

$- 2 x^{2} \sqrt[3]{9 x}$