Basic Math Examples

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

Step 1

Expand $(2 \sqrt[3]{3} + \sqrt[3]{2}) (\sqrt[3]{3} - 2 \sqrt[3]{2})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $2 \sqrt[3]{3} \sqrt[3]{3}$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

Add $1$ and $1$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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Step 2.1.4.1

Multiply $- 2$ by $2$ .

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

Step 2.1.4.2

Combine using the product rule for radicals.

$2 \sqrt[3]{9} - 4 \sqrt[3]{2 \cdot 3} + \sqrt[3]{2} \sqrt[3]{3} + \sqrt[3]{2} (- 2 \sqrt[3]{2})$

Step 2.1.4.3

Multiply $2$ by $3$ .

$2 \sqrt[3]{9} - 4 \sqrt[3]{6} + \sqrt[3]{2} \sqrt[3]{3} + \sqrt[3]{2} (- 2 \sqrt[3]{2})$

Step 2.1.5

Combine using the product rule for radicals.

$2 \sqrt[3]{9} - 4 \sqrt[3]{6} + \sqrt[3]{2 \cdot 3} + \sqrt[3]{2} (- 2 \sqrt[3]{2})$

Step 2.1.6

Multiply $2$ by $3$ .

$2 \sqrt[3]{9} - 4 \sqrt[3]{6} + \sqrt[3]{6} + \sqrt[3]{2} (- 2 \sqrt[3]{2})$

Step 2.1.7

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

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Step 2.1.7.1

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

$2 \sqrt[3]{9} - 4 \sqrt[3]{6} + \sqrt[3]{6} - 2 ({\sqrt[3]{2}}^{1} \sqrt[3]{2})$

Step 2.1.7.2

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

$2 \sqrt[3]{9} - 4 \sqrt[3]{6} + \sqrt[3]{6} - 2 ({\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{1})$

Step 2.1.7.3

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

$2 \sqrt[3]{9} - 4 \sqrt[3]{6} + \sqrt[3]{6} - 2 {\sqrt[3]{2}}^{1 + 1}$

Step 2.1.7.4

Add $1$ and $1$ .

$2 \sqrt[3]{9} - 4 \sqrt[3]{6} + \sqrt[3]{6} - 2 {\sqrt[3]{2}}^{2}$

Step 2.1.8

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

$2 \sqrt[3]{9} - 4 \sqrt[3]{6} + \sqrt[3]{6} - 2 \sqrt[3]{2^{2}}$

Step 2.1.9

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

$2 \sqrt[3]{9} - 4 \sqrt[3]{6} + \sqrt[3]{6} - 2 \sqrt[3]{4}$

Step 2.2

Add $- 4 \sqrt[3]{6}$ and $\sqrt[3]{6}$ .

$2 \sqrt[3]{9} - 3 \sqrt[3]{6} - 2 \sqrt[3]{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt[3]{9} - 3 \sqrt[3]{6} - 2 \sqrt[3]{4}$

Decimal Form:

$- 4.46599623 \dots$