Algebra Examples

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

Step 1

Rewrite ${(\frac{x}{2} + \frac{y}{3})}^{2}$ as $(\frac{x}{2} + \frac{y}{3}) (\frac{x}{2} + \frac{y}{3})$ .

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

Step 2

Expand $(\frac{x}{2} + \frac{y}{3}) (\frac{x}{2} + \frac{y}{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

$\frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{y}{3} + \frac{y}{3} \cdot \frac{x}{2} + \frac{y}{3} \cdot \frac{y}{3}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\frac{x}{2} \cdot \frac{x}{2}$ .

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Step 3.1.1.1

Multiply $\frac{x}{2}$ by $\frac{x}{2}$ .

$\frac{x \cdot x}{2 \cdot 2} + \frac{x}{2} \cdot \frac{y}{3} + \frac{y}{3} \cdot \frac{x}{2} + \frac{y}{3} \cdot \frac{y}{3}$

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

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

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

Step 3.1.1.5

Add $1$ and $1$ .

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

Step 3.1.1.6

Multiply $2$ by $2$ .

$\frac{x^{2}}{4} + \frac{x}{2} \cdot \frac{y}{3} + \frac{y}{3} \cdot \frac{x}{2} + \frac{y}{3} \cdot \frac{y}{3}$

Step 3.1.2

Multiply $\frac{x}{2} \cdot \frac{y}{3}$ .

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Step 3.1.2.1

Multiply $\frac{x}{2}$ by $\frac{y}{3}$ .

$\frac{x^{2}}{4} + \frac{x y}{2 \cdot 3} + \frac{y}{3} \cdot \frac{x}{2} + \frac{y}{3} \cdot \frac{y}{3}$

Step 3.1.2.2

Multiply $2$ by $3$ .

$\frac{x^{2}}{4} + \frac{x y}{6} + \frac{y}{3} \cdot \frac{x}{2} + \frac{y}{3} \cdot \frac{y}{3}$

Step 3.1.3

Multiply $\frac{y}{3} \cdot \frac{x}{2}$ .

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Step 3.1.3.1

Multiply $\frac{y}{3}$ by $\frac{x}{2}$ .

$\frac{x^{2}}{4} + \frac{x y}{6} + \frac{y x}{3 \cdot 2} + \frac{y}{3} \cdot \frac{y}{3}$

Step 3.1.3.2

Multiply $3$ by $2$ .

$\frac{x^{2}}{4} + \frac{x y}{6} + \frac{y x}{6} + \frac{y}{3} \cdot \frac{y}{3}$

Step 3.1.4

Multiply $\frac{y}{3} \cdot \frac{y}{3}$ .

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Step 3.1.4.1

Multiply $\frac{y}{3}$ by $\frac{y}{3}$ .

$\frac{x^{2}}{4} + \frac{x y}{6} + \frac{y x}{6} + \frac{y \cdot y}{3 \cdot 3}$

Step 3.1.4.2

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

$\frac{x^{2}}{4} + \frac{x y}{6} + \frac{y x}{6} + \frac{y^{1} y}{3 \cdot 3}$

Step 3.1.4.3

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

$\frac{x^{2}}{4} + \frac{x y}{6} + \frac{y x}{6} + \frac{y^{1} y^{1}}{3 \cdot 3}$

Step 3.1.4.4

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

$\frac{x^{2}}{4} + \frac{x y}{6} + \frac{y x}{6} + \frac{y^{1 + 1}}{3 \cdot 3}$

Step 3.1.4.5

Add $1$ and $1$ .

$\frac{x^{2}}{4} + \frac{x y}{6} + \frac{y x}{6} + \frac{y^{2}}{3 \cdot 3}$

Step 3.1.4.6

Multiply $3$ by $3$ .

$\frac{x^{2}}{4} + \frac{x y}{6} + \frac{y x}{6} + \frac{y^{2}}{9}$

Step 3.2

Add $\frac{x y}{6}$ and $\frac{y x}{6}$ .

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Step 3.2.1

Reorder $y$ and $x$ .

$\frac{x^{2}}{4} + \frac{x y}{6} + \frac{x y}{6} + \frac{y^{2}}{9}$

Step 3.2.2

Add $\frac{x y}{6}$ and $\frac{x y}{6}$ .

$\frac{x^{2}}{4} + 2 \frac{x y}{6} + \frac{y^{2}}{9}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $6$ .

$\frac{x^{2}}{4} + 2 \frac{x y}{2 (3)} + \frac{y^{2}}{9}$

Step 4.2

Cancel the common factor.

$\frac{x^{2}}{4} + 2 \frac{x y}{2 \cdot 3} + \frac{y^{2}}{9}$

Step 4.3

Rewrite the expression.

$\frac{x^{2}}{4} + \frac{x y}{3} + \frac{y^{2}}{9}$