Algebra Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

To write $\frac{4}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 2.2

To write $- \frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 2.3

Write each expression with a common denominator of $y x$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

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

Step 2.3.2

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

$(16 x^{2} - y^{2}) \frac{\frac{4 x}{y x} - \frac{y}{x y}}{x y}$

Step 2.3.3

Reorder the factors of $y x$ .

$(16 x^{2} - y^{2}) \frac{\frac{4 x}{x y} - \frac{y}{x y}}{x y}$

Step 2.4

Combine the numerators over the common denominator.

$(16 x^{2} - y^{2}) \frac{\frac{4 x - y}{x y}}{x y}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Multiply $\frac{4 x - y}{x y} \cdot \frac{1}{x y}$ .

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Step 4.1

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

$(16 x^{2} - y^{2}) \frac{4 x - y}{x y (x y)}$

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

$(16 x^{2} - y^{2}) \frac{4 x - y}{x^{2} y \cdot y}$

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

Add $1$ and $1$ .

$(16 x^{2} - y^{2}) \frac{4 x - y}{x^{2} y^{2}}$

Step 5

Multiply $16 x^{2} - y^{2}$ by $\frac{4 x - y}{x^{2} y^{2}}$ .

$\frac{(16 x^{2} - y^{2}) (4 x - y)}{x^{2} y^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

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

Step 6.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 x$ and $b = y$ .

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

Step 6.3

Combine exponents.

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Step 6.3.1

Raise $4 x - y$ to the power of $1$ .

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

Step 6.3.2

Raise $4 x - y$ to the power of $1$ .

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

Step 6.3.3

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

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

Step 6.3.4

Add $1$ and $1$ .

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