Basic Math Examples

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

Step 1

Expand $(\sqrt[3]{a} - \sqrt[3]{b}) (\sqrt[3]{a^{2}} + \sqrt[3]{a b} + \sqrt[3]{b^{2}})$ by multiplying each term in the first expression by each term in the second expression.

$\sqrt[3]{a} \sqrt[3]{a^{2}} + \sqrt[3]{a} \sqrt[3]{a b} + \sqrt[3]{a} \sqrt[3]{b^{2}} - \sqrt[3]{b} \sqrt[3]{a^{2}} - \sqrt[3]{b} \sqrt[3]{a b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

Combine using the product rule for radicals.

$\sqrt[3]{a \cdot a^{2}} + \sqrt[3]{a} \sqrt[3]{a b} + \sqrt[3]{a} \sqrt[3]{b^{2}} - \sqrt[3]{b} \sqrt[3]{a^{2}} - \sqrt[3]{b} \sqrt[3]{a b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

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Step 2.1.1.2.1.1

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

$\sqrt[3]{a^{1} a^{2}} + \sqrt[3]{a} \sqrt[3]{a b} + \sqrt[3]{a} \sqrt[3]{b^{2}} - \sqrt[3]{b} \sqrt[3]{a^{2}} - \sqrt[3]{b} \sqrt[3]{a b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.1.2.1.2

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

$\sqrt[3]{a^{1 + 2}} + \sqrt[3]{a} \sqrt[3]{a b} + \sqrt[3]{a} \sqrt[3]{b^{2}} - \sqrt[3]{b} \sqrt[3]{a^{2}} - \sqrt[3]{b} \sqrt[3]{a b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.1.2.2

Add $1$ and $2$ .

$\sqrt[3]{a^{3}} + \sqrt[3]{a} \sqrt[3]{a b} + \sqrt[3]{a} \sqrt[3]{b^{2}} - \sqrt[3]{b} \sqrt[3]{a^{2}} - \sqrt[3]{b} \sqrt[3]{a b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.2

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

$a + \sqrt[3]{a} \sqrt[3]{a b} + \sqrt[3]{a} \sqrt[3]{b^{2}} - \sqrt[3]{b} \sqrt[3]{a^{2}} - \sqrt[3]{b} \sqrt[3]{a b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.3

Multiply $\sqrt[3]{a} \sqrt[3]{a b}$ .

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Step 2.1.3.1

Combine using the product rule for radicals.

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

Step 2.1.3.2

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

$a + \sqrt[3]{a^{1} a b} + \sqrt[3]{a} \sqrt[3]{b^{2}} - \sqrt[3]{b} \sqrt[3]{a^{2}} - \sqrt[3]{b} \sqrt[3]{a b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.3.3

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

$a + \sqrt[3]{a^{1} a^{1} b} + \sqrt[3]{a} \sqrt[3]{b^{2}} - \sqrt[3]{b} \sqrt[3]{a^{2}} - \sqrt[3]{b} \sqrt[3]{a b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.3.4

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

$a + \sqrt[3]{a^{1 + 1} b} + \sqrt[3]{a} \sqrt[3]{b^{2}} - \sqrt[3]{b} \sqrt[3]{a^{2}} - \sqrt[3]{b} \sqrt[3]{a b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.3.5

Add $1$ and $1$ .

$a + \sqrt[3]{a^{2} b} + \sqrt[3]{a} \sqrt[3]{b^{2}} - \sqrt[3]{b} \sqrt[3]{a^{2}} - \sqrt[3]{b} \sqrt[3]{a b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.4

Combine using the product rule for radicals.

$a + \sqrt[3]{a^{2} b} + \sqrt[3]{a b^{2}} - \sqrt[3]{b} \sqrt[3]{a^{2}} - \sqrt[3]{b} \sqrt[3]{a b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.5

Combine using the product rule for radicals.

$a + \sqrt[3]{a^{2} b} + \sqrt[3]{a b^{2}} - \sqrt[3]{a^{2} b} - \sqrt[3]{b} \sqrt[3]{a b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.6

Multiply $- \sqrt[3]{b} \sqrt[3]{a b}$ .

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Step 2.1.6.1

Combine using the product rule for radicals.

$a + \sqrt[3]{a^{2} b} + \sqrt[3]{a b^{2}} - \sqrt[3]{a^{2} b} - \sqrt[3]{a b \cdot b} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.6.2

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

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

Step 2.1.6.3

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

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

Step 2.1.6.4

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

$a + \sqrt[3]{a^{2} b} + \sqrt[3]{a b^{2}} - \sqrt[3]{a^{2} b} - \sqrt[3]{a b^{1 + 1}} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.6.5

Add $1$ and $1$ .

$a + \sqrt[3]{a^{2} b} + \sqrt[3]{a b^{2}} - \sqrt[3]{a^{2} b} - \sqrt[3]{a b^{2}} - \sqrt[3]{b} \sqrt[3]{b^{2}}$

Step 2.1.7

Multiply $- \sqrt[3]{b} \sqrt[3]{b^{2}}$ .

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Step 2.1.7.1

Combine using the product rule for radicals.

$a + \sqrt[3]{a^{2} b} + \sqrt[3]{a b^{2}} - \sqrt[3]{a^{2} b} - \sqrt[3]{a b^{2}} - \sqrt[3]{b^{2} b}$

Step 2.1.7.2

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

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Step 2.1.7.2.1

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

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Step 2.1.7.2.1.1

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

$a + \sqrt[3]{a^{2} b} + \sqrt[3]{a b^{2}} - \sqrt[3]{a^{2} b} - \sqrt[3]{a b^{2}} - \sqrt[3]{b^{2} b^{1}}$

Step 2.1.7.2.1.2

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

$a + \sqrt[3]{a^{2} b} + \sqrt[3]{a b^{2}} - \sqrt[3]{a^{2} b} - \sqrt[3]{a b^{2}} - \sqrt[3]{b^{2 + 1}}$

Step 2.1.7.2.2

Add $2$ and $1$ .

$a + \sqrt[3]{a^{2} b} + \sqrt[3]{a b^{2}} - \sqrt[3]{a^{2} b} - \sqrt[3]{a b^{2}} - \sqrt[3]{b^{3}}$

Step 2.1.8

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

$a + \sqrt[3]{a^{2} b} + \sqrt[3]{a b^{2}} - \sqrt[3]{a^{2} b} - \sqrt[3]{a b^{2}} - b$

Step 2.2

Combine the opposite terms in $a + \sqrt[3]{a^{2} b} + \sqrt[3]{a b^{2}} - \sqrt[3]{a^{2} b} - \sqrt[3]{a b^{2}} - b$ .

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Step 2.2.1

Subtract $\sqrt[3]{a^{2} b}$ from $\sqrt[3]{a^{2} b}$ .

$a + 0 + \sqrt[3]{a b^{2}} - \sqrt[3]{a b^{2}} - b$

Step 2.2.2

Add $a$ and $0$ .

$a + \sqrt[3]{a b^{2}} - \sqrt[3]{a b^{2}} - b$

Step 2.2.3

Subtract $\sqrt[3]{a b^{2}}$ from $\sqrt[3]{a b^{2}}$ .

$a + 0 - b$

Step 2.2.4

Add $a$ and $0$ .

$a - b$