Finite Math Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

Combine using the product rule for radicals.

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

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

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Step 2.1.1.2.1.1

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

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

Step 2.1.1.2.1.2

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

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

Step 2.1.1.2.2

Add $1$ and $2$ .

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

Step 2.1.2

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

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

Step 2.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.1.4

Multiply $- \sqrt[3]{x} \sqrt[3]{x y}$ .

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Step 2.1.4.1

Combine using the product rule for radicals.

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

Step 2.1.4.2

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

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

Step 2.1.4.3

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

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

Step 2.1.4.4

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

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

Step 2.1.4.5

Add $1$ and $1$ .

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

Step 2.1.5

Combine using the product rule for radicals.

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

Step 2.1.6

Combine using the product rule for radicals.

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

Step 2.1.7

Rewrite using the commutative property of multiplication.

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

Step 2.1.8

Multiply $- \sqrt[3]{y} \sqrt[3]{x y}$ .

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Step 2.1.8.1

Combine using the product rule for radicals.

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

Step 2.1.8.2

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

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

Step 2.1.8.3

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

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

Step 2.1.8.4

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

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

Step 2.1.8.5

Add $1$ and $1$ .

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

Step 2.1.9

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

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Step 2.1.9.1

Combine using the product rule for radicals.

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

Step 2.1.9.2

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

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Step 2.1.9.2.1

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

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Step 2.1.9.2.1.1

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

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

Step 2.1.9.2.1.2

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

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

Step 2.1.9.2.2

Add $1$ and $2$ .

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

Step 2.1.10

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

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

Step 2.2

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

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Step 2.2.1

Subtract $\sqrt[3]{x y^{2}}$ from $\sqrt[3]{x y^{2}}$ .

$x - \sqrt[3]{x^{2} y} + 0 + \sqrt[3]{y x^{2}} + y$

Step 2.2.2

Add $x - \sqrt[3]{x^{2} y}$ and $0$ .

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

Step 3

Add $- \sqrt[3]{x^{2} y}$ and $\sqrt[3]{y x^{2}}$ .

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Step 3.1

Reorder $y$ and $x^{2}$ .

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

Step 3.2

Add $- \sqrt[3]{x^{2} y}$ and $\sqrt[3]{x^{2} y}$ .

$x + 0 + y$

Step 4

Add $x$ and $0$ .

$x + y$