Precalculus Examples

$(x^{\frac{1}{3}} + y^{\frac{1}{3}}) (x^{\frac{2}{3}} + x^{\frac{1}{3}} y^{\frac{1}{3}} + y^{\frac{2}{3}})$

Step 1

Expand $(x^{\frac{1}{3}} + y^{\frac{1}{3}}) (x^{\frac{2}{3}} + x^{\frac{1}{3}} y^{\frac{1}{3}} + y^{\frac{2}{3}})$ by multiplying each term in the first expression by each term in the second expression.

$x^{\frac{1}{3}} x^{\frac{2}{3}} + x^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2

Simplify each term.

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Step 2.1

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 2.1.1

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

$x^{\frac{1}{3} + \frac{2}{3}} + x^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.1.2

Combine the numerators over the common denominator.

$x^{\frac{1 + 2}{3}} + x^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.1.3

Add $1$ and $2$ .

$x^{\frac{3}{3}} + x^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.1.4

Divide $3$ by $3$ .

$x^{1} + x^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.2

Simplify $x^{1}$ .

$x + x^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.3

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 2.3.1

Move $x^{\frac{1}{3}}$ .

$x + x^{\frac{1}{3}} x^{\frac{1}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.3.2

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

$x + x^{\frac{1}{3} + \frac{1}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.3.3

Combine the numerators over the common denominator.

$x + x^{\frac{1 + 1}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.3.4

Add $1$ and $1$ .

$x + x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1}{3}} (x^{\frac{1}{3}} y^{\frac{1}{3}}) + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.4

Multiply $y^{\frac{1}{3}}$ by $y^{\frac{1}{3}}$ by adding the exponents.

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Step 2.4.1

Move $y^{\frac{1}{3}}$ .

$x + x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1}{3}} y^{\frac{1}{3}} x^{\frac{1}{3}} + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.4.2

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

$x + x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1}{3} + \frac{1}{3}} x^{\frac{1}{3}} + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.4.3

Combine the numerators over the common denominator.

$x + x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{1 + 1}{3}} x^{\frac{1}{3}} + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.4.4

Add $1$ and $1$ .

$x + x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{2}{3}} x^{\frac{1}{3}} + y^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 2.5

Multiply $y^{\frac{1}{3}}$ by $y^{\frac{2}{3}}$ by adding the exponents.

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Step 2.5.1

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

$x + x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{2}{3}} x^{\frac{1}{3}} + y^{\frac{1}{3} + \frac{2}{3}}$

Step 2.5.2

Combine the numerators over the common denominator.

$x + x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{2}{3}} x^{\frac{1}{3}} + y^{\frac{1 + 2}{3}}$

Step 2.5.3

Add $1$ and $2$ .

$x + x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{2}{3}} x^{\frac{1}{3}} + y^{\frac{3}{3}}$

Step 2.5.4

Divide $3$ by $3$ .

$x + x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{2}{3}} x^{\frac{1}{3}} + y^{1}$

Step 2.6

Simplify $y^{1}$ .

$x + x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} x^{\frac{2}{3}} + y^{\frac{2}{3}} x^{\frac{1}{3}} + y$

Step 3

Add $x^{\frac{2}{3}} y^{\frac{1}{3}}$ and $y^{\frac{1}{3}} x^{\frac{2}{3}}$ .

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Step 3.1

Reorder $y^{\frac{1}{3}}$ and $x^{\frac{2}{3}}$ .

$x + x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{2}{3}} x^{\frac{1}{3}} + y$

Step 3.2

Add $x^{\frac{2}{3}} y^{\frac{1}{3}}$ and $x^{\frac{2}{3}} y^{\frac{1}{3}}$ .

$x + 2 x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{2}{3}} x^{\frac{1}{3}} + y$

Step 4

Add $x^{\frac{1}{3}} y^{\frac{2}{3}}$ and $y^{\frac{2}{3}} x^{\frac{1}{3}}$ .

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Step 4.1

Reorder $y^{\frac{2}{3}}$ and $x^{\frac{1}{3}}$ .

$x + 2 x^{\frac{2}{3}} y^{\frac{1}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + x^{\frac{1}{3}} y^{\frac{2}{3}} + y$

Step 4.2

Add $x^{\frac{1}{3}} y^{\frac{2}{3}}$ and $x^{\frac{1}{3}} y^{\frac{2}{3}}$ .

$x + 2 x^{\frac{2}{3}} y^{\frac{1}{3}} + 2 x^{\frac{1}{3}} y^{\frac{2}{3}} + y$

Step 5

Move $2 x^{\frac{1}{3}} y^{\frac{2}{3}}$ .

$x + 2 x^{\frac{2}{3}} y^{\frac{1}{3}} + y + 2 x^{\frac{1}{3}} y^{\frac{2}{3}}$

Step 6

Move $2 x^{\frac{2}{3}} y^{\frac{1}{3}}$ .

$x + y + 2 x^{\frac{2}{3}} y^{\frac{1}{3}} + 2 x^{\frac{1}{3}} y^{\frac{2}{3}}$