Algebra Examples

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

Step 1

Combine fractions.

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Step 1.1

Combine $\frac{1}{3}$ and $x$ .

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

Step 1.2

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

$(\frac{x}{3} - 3 y) (\frac{x}{3} - 3 y)$

Step 2

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

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Step 2.1

Apply the distributive property.

$\frac{x}{3} (\frac{x}{3} - 3 y) - 3 y (\frac{x}{3} - 3 y)$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $\frac{x}{3} \cdot \frac{x}{3}$ .

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

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

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

Step 3.1.1.5

Add $1$ and $1$ .

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

Step 3.1.1.6

Multiply $3$ by $3$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

Cancel the common factor of $3$ .

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Step 3.1.3.1

Factor $3$ out of $- 3$ .

$\frac{x^{2}}{9} + 3 (- 1) \frac{x}{3} y - 3 y \frac{x}{3} - 3 y (- 3 y)$

Step 3.1.3.2

Cancel the common factor.

$\frac{x^{2}}{9} + 3 \cdot - 1 \frac{x}{3} y - 3 y \frac{x}{3} - 3 y (- 3 y)$

Step 3.1.3.3

Rewrite the expression.

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

Step 3.1.4

Cancel the common factor of $3$ .

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Step 3.1.4.1

Factor $3$ out of $- 3 y$ .

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

Step 3.1.4.2

Cancel the common factor.

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

Step 3.1.4.3

Rewrite the expression.

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

Step 3.1.5

Rewrite using the commutative property of multiplication.

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

Step 3.1.6

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.6.1

Move $y$ .

$\frac{x^{2}}{9} - x y - y x - 3 \cdot - 3 (y \cdot y)$

Step 3.1.6.2

Multiply $y$ by $y$ .

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

Step 3.1.7

Multiply $- 3$ by $- 3$ .

$\frac{x^{2}}{9} - x y - y x + 9 y^{2}$

Step 3.2

Subtract $y x$ from $- x y$ .

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Step 3.2.1

Move $y$ .

$\frac{x^{2}}{9} - x y - 1 x y + 9 y^{2}$

Step 3.2.2

Subtract $x y$ from $- x y$ .

$\frac{x^{2}}{9} - 2 x y + 9 y^{2}$