Algebra Examples

${(\frac{2}{3 x} - \frac{1}{4 y})}^{2}$

Step 1

Rewrite ${(\frac{2}{3 x} - \frac{1}{4 y})}^{2}$ as $(\frac{2}{3 x} - \frac{1}{4 y}) (\frac{2}{3 x} - \frac{1}{4 y})$ .

$(\frac{2}{3 x} - \frac{1}{4 y}) (\frac{2}{3 x} - \frac{1}{4 y})$

Step 2

Expand $(\frac{2}{3 x} - \frac{1}{4 y}) (\frac{2}{3 x} - \frac{1}{4 y})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{2}{3 x} (\frac{2}{3 x} - \frac{1}{4 y}) - \frac{1}{4 y} (\frac{2}{3 x} - \frac{1}{4 y})$

Step 2.2

Apply the distributive property.

$\frac{2}{3 x} \cdot \frac{2}{3 x} + \frac{2}{3 x} (- \frac{1}{4 y}) - \frac{1}{4 y} (\frac{2}{3 x} - \frac{1}{4 y})$

Step 2.3

Apply the distributive property.

$\frac{2}{3 x} \cdot \frac{2}{3 x} + \frac{2}{3 x} (- \frac{1}{4 y}) - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

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

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Step 3.1.1.1

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

$\frac{2 \cdot 2}{3 x (3 x)} + \frac{2}{3 x} (- \frac{1}{4 y}) - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.1.2

Multiply $2$ by $2$ .

$\frac{4}{3 x (3 x)} + \frac{2}{3 x} (- \frac{1}{4 y}) - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.1.3

Multiply $3$ by $3$ .

$\frac{4}{9 x \cdot x} + \frac{2}{3 x} (- \frac{1}{4 y}) - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.1.4

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

$\frac{4}{9 (x^{1} x)} + \frac{2}{3 x} (- \frac{1}{4 y}) - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.1.5

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

$\frac{4}{9 (x^{1} x^{1})} + \frac{2}{3 x} (- \frac{1}{4 y}) - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.1.6

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

$\frac{4}{9 x^{1 + 1}} + \frac{2}{3 x} (- \frac{1}{4 y}) - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.1.7

Add $1$ and $1$ .

$\frac{4}{9 x^{2}} + \frac{2}{3 x} (- \frac{1}{4 y}) - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.2

Rewrite using the commutative property of multiplication.

$\frac{4}{9 x^{2}} - \frac{2}{3 x} \cdot \frac{1}{4 y} - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.3

Cancel the common factor of $2$ .

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Step 3.1.3.1

Move the leading negative in $- \frac{2}{3 x}$ into the numerator.

$\frac{4}{9 x^{2}} + \frac{- 2}{3 x} \cdot \frac{1}{4 y} - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.3.2

Factor $2$ out of $- 2$ .

$\frac{4}{9 x^{2}} + \frac{2 (- 1)}{3 x} \cdot \frac{1}{4 y} - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.3.3

Factor $2$ out of $4 y$ .

$\frac{4}{9 x^{2}} + \frac{2 (- 1)}{3 x} \cdot \frac{1}{2 (2 y)} - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.3.4

Cancel the common factor.

$\frac{4}{9 x^{2}} + \frac{2 \cdot - 1}{3 x} \cdot \frac{1}{2 (2 y)} - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.3.5

Rewrite the expression.

$\frac{4}{9 x^{2}} + \frac{- 1}{3 x} \cdot \frac{1}{2 y} - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.4

Multiply $\frac{- 1}{3 x}$ by $\frac{1}{2 y}$ .

$\frac{4}{9 x^{2}} + \frac{- 1}{3 x (2 y)} - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.5

Multiply $2$ by $3$ .

$\frac{4}{9 x^{2}} + \frac{- 1}{6 x y} - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.6

Move the negative in front of the fraction.

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} - \frac{1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.7

Cancel the common factor of $2$ .

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Step 3.1.7.1

Move the leading negative in $- \frac{1}{4 y}$ into the numerator.

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} + \frac{- 1}{4 y} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.7.2

Factor $2$ out of $4 y$ .

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} + \frac{- 1}{2 (2 y)} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.7.3

Cancel the common factor.

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} + \frac{- 1}{2 (2 y)} \cdot \frac{2}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.7.4

Rewrite the expression.

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} + \frac{- 1}{2 y} \cdot \frac{1}{3 x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.8

Multiply $\frac{- 1}{2 y}$ by $\frac{1}{3 x}$ .

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} + \frac{- 1}{2 y (3 x)} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.9

Multiply $3$ by $2$ .

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} + \frac{- 1}{6 y x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.10

Move the negative in front of the fraction.

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} - \frac{1}{6 y x} - \frac{1}{4 y} (- \frac{1}{4 y})$

Step 3.1.11

Multiply $- \frac{1}{4 y} (- \frac{1}{4 y})$ .

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Step 3.1.11.1

Multiply $- 1$ by $- 1$ .

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} - \frac{1}{6 y x} + 1 \frac{1}{4 y} \frac{1}{4 y}$

Step 3.1.11.2

Multiply $\frac{1}{4 y}$ by $1$ .

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} - \frac{1}{6 y x} + \frac{1}{4 y} \cdot \frac{1}{4 y}$

Step 3.1.11.3

Multiply $\frac{1}{4 y}$ by $\frac{1}{4 y}$ .

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} - \frac{1}{6 y x} + \frac{1}{4 y (4 y)}$

Step 3.1.11.4

Multiply $4$ by $4$ .

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} - \frac{1}{6 y x} + \frac{1}{16 y \cdot y}$

Step 3.1.11.5

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

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} - \frac{1}{6 y x} + \frac{1}{16 (y^{1} y)}$

Step 3.1.11.6

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

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} - \frac{1}{6 y x} + \frac{1}{16 (y^{1} y^{1})}$

Step 3.1.11.7

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

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} - \frac{1}{6 y x} + \frac{1}{16 y^{1 + 1}}$

Step 3.1.11.8

Add $1$ and $1$ .

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} - \frac{1}{6 y x} + \frac{1}{16 y^{2}}$

Step 3.2

Subtract $\frac{1}{6 y x}$ from $- \frac{1}{6 x y}$ .

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Step 3.2.1

Move $y$ .

$\frac{4}{9 x^{2}} - \frac{1}{6 x y} - \frac{1}{6 x y} + \frac{1}{16 y^{2}}$

Step 3.2.2

Subtract $\frac{1}{6 x y}$ from $- \frac{1}{6 x y}$ .

$\frac{4}{9 x^{2}} - 2 \frac{1}{6 x y} + \frac{1}{16 y^{2}}$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Factor $2$ out of $- 2$ .

$\frac{4}{9 x^{2}} + 2 (- 1) \frac{1}{6 x y} + \frac{1}{16 y^{2}}$

Step 4.1.2

Factor $2$ out of $6 x y$ .

$\frac{4}{9 x^{2}} + 2 (- 1) \frac{1}{2 (3 x y)} + \frac{1}{16 y^{2}}$

Step 4.1.3

Cancel the common factor.

$\frac{4}{9 x^{2}} + 2 \cdot - 1 \frac{1}{2 (3 x y)} + \frac{1}{16 y^{2}}$

Step 4.1.4

Rewrite the expression.

$\frac{4}{9 x^{2}} - 1 \frac{1}{3 x y} + \frac{1}{16 y^{2}}$

Step 4.2

Rewrite $- 1 \frac{1}{3 x y}$ as $- \frac{1}{3 x y}$ .

$\frac{4}{9 x^{2}} - \frac{1}{3 x y} + \frac{1}{16 y^{2}}$