Algebra Examples

$\sqrt[3]{4 x^{2}} (\sqrt[3]{2 x} - 5 \sqrt[3]{4 x^{2}})$

Step 1

Apply the distributive property.

$\sqrt[3]{4 x^{2}} \sqrt[3]{2 x} + \sqrt[3]{4 x^{2}} (- 5 \sqrt[3]{4 x^{2}})$

Step 2

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

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Step 2.1

Combine using the product rule for radicals.

$\sqrt[3]{4 x^{2} (2 x)} + \sqrt[3]{4 x^{2}} (- 5 \sqrt[3]{4 x^{2}})$

Step 2.2

Multiply $2$ by $4$ .

$\sqrt[3]{8 x^{2} x} + \sqrt[3]{4 x^{2}} (- 5 \sqrt[3]{4 x^{2}})$

Step 2.3

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

$\sqrt[3]{8 (x^{1} x^{2})} + \sqrt[3]{4 x^{2}} (- 5 \sqrt[3]{4 x^{2}})$

Step 2.4

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

$\sqrt[3]{8 x^{1 + 2}} + \sqrt[3]{4 x^{2}} (- 5 \sqrt[3]{4 x^{2}})$

Step 2.5

Add $1$ and $2$ .

$\sqrt[3]{8 x^{3}} + \sqrt[3]{4 x^{2}} (- 5 \sqrt[3]{4 x^{2}})$

Step 3

Rewrite using the commutative property of multiplication.

$\sqrt[3]{8 x^{3}} - 5 \sqrt[3]{4 x^{2}} \sqrt[3]{4 x^{2}}$

Step 4

Simplify each term.

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Step 4.1

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

$\sqrt[3]{{(2 x)}^{3}} - 5 \sqrt[3]{4 x^{2}} \sqrt[3]{4 x^{2}}$

Step 4.2

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

$2 x - 5 \sqrt[3]{4 x^{2}} \sqrt[3]{4 x^{2}}$

Step 4.3

Multiply $- 5 \sqrt[3]{4 x^{2}} \sqrt[3]{4 x^{2}}$ .

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Step 4.3.1

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

$2 x - 5 ({\sqrt[3]{4 x^{2}}}^{1} \sqrt[3]{4 x^{2}})$

Step 4.3.2

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

$2 x - 5 ({\sqrt[3]{4 x^{2}}}^{1} {\sqrt[3]{4 x^{2}}}^{1})$

Step 4.3.3

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

$2 x - 5 {\sqrt[3]{4 x^{2}}}^{1 + 1}$

Step 4.3.4

Add $1$ and $1$ .

$2 x - 5 {\sqrt[3]{4 x^{2}}}^{2}$

Step 4.4

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

$2 x - 5 \sqrt[3]{{(4 x^{2})}^{2}}$

Step 4.5

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

$2 x - 5 \sqrt[3]{4^{2} {(x^{2})}^{2}}$

Step 4.6

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

$2 x - 5 \sqrt[3]{16 {(x^{2})}^{2}}$

Step 4.7

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

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Step 4.7.1

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

$2 x - 5 \sqrt[3]{16 x^{2 \cdot 2}}$

Step 4.7.2

Multiply $2$ by $2$ .

$2 x - 5 \sqrt[3]{16 x^{4}}$

Step 4.8

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

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Step 4.8.1

Factor $8$ out of $16$ .

$2 x - 5 \sqrt[3]{8 (2) x^{4}}$

Step 4.8.2

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

$2 x - 5 \sqrt[3]{2^{3} \cdot 2 x^{4}}$

Step 4.8.3

Factor out $x^{3}$ .

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

Step 4.8.4

Move $2$ .

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

Step 4.8.5

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

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

Step 4.8.6

Add parentheses.

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

Step 4.9

Pull terms out from under the radical.

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

Step 4.10

Multiply $2$ by $- 5$ .

$2 x - 10 x \sqrt[3]{2 x}$