Algebra Examples

$4 x \sqrt[3]{4 x^{2}} (2 \sqrt[3]{32 x^{2}} - x \sqrt[3]{2 x})$

Step 1

Simplify each term.

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Step 1.1

Rewrite $32 x^{2}$ as $2^{3} \cdot (4 x^{2})$ .

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Step 1.1.1

Factor $8$ out of $32$ .

$4 x \sqrt[3]{4 x^{2}} (2 \sqrt[3]{8 (4) x^{2}} - x \sqrt[3]{2 x})$

Step 1.1.2

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

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

Step 1.1.3

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Multiply $2$ by $2$ .

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

Step 2

Apply the distributive property.

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

Step 3

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

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Step 3.1

Multiply $4$ by $4$ .

$16 x \sqrt[3]{4 x^{2}} \sqrt[3]{4 x^{2}} + 4 x \sqrt[3]{4 x^{2}} (- x \sqrt[3]{2 x})$

Step 3.2

Raise $\sqrt[3]{4 x^{2}}$ to the power of $1$ .

$16 x ({\sqrt[3]{4 x^{2}}}^{1} \sqrt[3]{4 x^{2}}) + 4 x \sqrt[3]{4 x^{2}} (- x \sqrt[3]{2 x})$

Step 3.3

Raise $\sqrt[3]{4 x^{2}}$ to the power of $1$ .

$16 x ({\sqrt[3]{4 x^{2}}}^{1} {\sqrt[3]{4 x^{2}}}^{1}) + 4 x \sqrt[3]{4 x^{2}} (- x \sqrt[3]{2 x})$

Step 3.4

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

$16 x {\sqrt[3]{4 x^{2}}}^{1 + 1} + 4 x \sqrt[3]{4 x^{2}} (- x \sqrt[3]{2 x})$

Step 3.5

Add $1$ and $1$ .

$16 x {\sqrt[3]{4 x^{2}}}^{2} + 4 x \sqrt[3]{4 x^{2}} (- x \sqrt[3]{2 x})$

Step 4

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

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Step 4.1

Move $x$ .

$16 x {\sqrt[3]{4 x^{2}}}^{2} + 4 (x \cdot x) \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 4.2

Multiply $x$ by $x$ .

$16 x {\sqrt[3]{4 x^{2}}}^{2} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5

Simplify each term.

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Step 5.1

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

$16 x \sqrt[3]{{(4 x^{2})}^{2}} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.2

Apply the product rule to $4 x^{2}$ .

$16 x \sqrt[3]{4^{2} {(x^{2})}^{2}} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.3

Raise $4$ to the power of $2$ .

$16 x \sqrt[3]{16 {(x^{2})}^{2}} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.4

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

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Step 5.4.1

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

$16 x \sqrt[3]{16 x^{2 \cdot 2}} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.4.2

Multiply $2$ by $2$ .

$16 x \sqrt[3]{16 x^{4}} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.5

Rewrite $16 x^{4}$ as ${(2 x)}^{3} \cdot (2 x)$ .

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Step 5.5.1

Factor $8$ out of $16$ .

$16 x \sqrt[3]{8 (2) x^{4}} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.5.2

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

$16 x \sqrt[3]{2^{3} \cdot 2 x^{4}} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.5.3

Factor out $x^{3}$ .

$16 x \sqrt[3]{2^{3} \cdot 2 (x^{3} x)} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.5.4

Move $2$ .

$16 x \sqrt[3]{2^{3} x^{3} \cdot 2 x} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.5.5

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

$16 x \sqrt[3]{{(2 x)}^{3} \cdot 2 x} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.5.6

Add parentheses.

$16 x \sqrt[3]{{(2 x)}^{3} \cdot (2 x)} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.6

Pull terms out from under the radical.

$16 x (2 x \sqrt[3]{2 x}) + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.7

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

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Step 5.7.1

Move $x$ .

$16 (x \cdot x) (2 \sqrt[3]{2 x}) + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.7.2

Multiply $x$ by $x$ .

$16 x^{2} (2 \sqrt[3]{2 x}) + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.8

Multiply $2$ by $16$ .

$32 x^{2} \sqrt[3]{2 x} + 4 x^{2} \sqrt[3]{4 x^{2}} (- 1 \sqrt[3]{2 x})$

Step 5.9

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

$32 x^{2} \sqrt[3]{2 x} + 4 x^{2} \sqrt[3]{4 x^{2}} (- \sqrt[3]{2 x})$

Step 5.10

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

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Step 5.10.1

Multiply $- 1$ by $4$ .

$32 x^{2} \sqrt[3]{2 x} - 4 x^{2} \sqrt[3]{4 x^{2}} \sqrt[3]{2 x}$

Step 5.10.2

Combine using the product rule for radicals.

$32 x^{2} \sqrt[3]{2 x} - 4 x^{2} \sqrt[3]{2 x (4 x^{2})}$

Step 5.10.3

Multiply $4$ by $2$ .

$32 x^{2} \sqrt[3]{2 x} - 4 x^{2} \sqrt[3]{8 x \cdot x^{2}}$

Step 5.10.4

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

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Step 5.10.4.1

Move $x^{2}$ .

$32 x^{2} \sqrt[3]{2 x} - 4 x^{2} \sqrt[3]{8 (x^{2} x)}$

Step 5.10.4.2

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

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Step 5.10.4.2.1

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

$32 x^{2} \sqrt[3]{2 x} - 4 x^{2} \sqrt[3]{8 (x^{2} x^{1})}$

Step 5.10.4.2.2

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

$32 x^{2} \sqrt[3]{2 x} - 4 x^{2} \sqrt[3]{8 x^{2 + 1}}$

Step 5.10.4.3

Add $2$ and $1$ .

$32 x^{2} \sqrt[3]{2 x} - 4 x^{2} \sqrt[3]{8 x^{3}}$

Step 5.11

Rewrite $8 x^{3}$ as ${(2 x)}^{3}$ .

$32 x^{2} \sqrt[3]{2 x} - 4 x^{2} \sqrt[3]{{(2 x)}^{3}}$

Step 5.12

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

$32 x^{2} \sqrt[3]{2 x} - 4 x^{2} (2 x)$

Step 5.13

Rewrite using the commutative property of multiplication.

$32 x^{2} \sqrt[3]{2 x} - 4 \cdot 2 x^{2} x$

Step 5.14

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

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Step 5.14.1

Move $x$ .

$32 x^{2} \sqrt[3]{2 x} - 4 \cdot 2 (x \cdot x^{2})$

Step 5.14.2

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

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Step 5.14.2.1

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

$32 x^{2} \sqrt[3]{2 x} - 4 \cdot 2 (x^{1} x^{2})$

Step 5.14.2.2

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

$32 x^{2} \sqrt[3]{2 x} - 4 \cdot 2 x^{1 + 2}$

Step 5.14.3

Add $1$ and $2$ .

$32 x^{2} \sqrt[3]{2 x} - 4 \cdot 2 x^{3}$

Step 5.15

Multiply $- 4$ by $2$ .

$32 x^{2} \sqrt[3]{2 x} - 8 x^{3}$