Simplify ((x^2)/4)/((x^2)/4-4/x)

$\frac{\frac{x^{2}}{4}}{\frac{x^{2}}{4} - \frac{4}{x}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2}}{4} \cdot \frac{1}{\frac{x^{2}}{4} - \frac{4}{x}}$

Step 2

Simplify the denominator.

To write $\frac{x^{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x^{2}}{4} \cdot \frac{1}{\frac{x^{2}}{4} \cdot \frac{x}{x} - \frac{4}{x}}$

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

$\frac{x^{2}}{4} \cdot \frac{1}{\frac{x^{2}}{4} \cdot \frac{x}{x} - \frac{4}{x} \cdot \frac{4}{4}}$

Write each expression with a common denominator of $4 x$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{x^{2}}{4} \cdot \frac{1}{\frac{x^{2} x}{4 x} - \frac{4}{x} \cdot \frac{4}{4}}$

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

$\frac{x^{2}}{4} \cdot \frac{1}{\frac{x^{2} x}{4 x} - \frac{4 \cdot 4}{x \cdot 4}}$

Reorder the factors of $x \cdot 4$ .

$\frac{x^{2}}{4} \cdot \frac{1}{\frac{x^{2} x}{4 x} - \frac{4 \cdot 4}{4 x}}$

Combine the numerators over the common denominator.

$\frac{x^{2}}{4} \cdot \frac{1}{\frac{x^{2} x - 4 \cdot 4}{4 x}}$

Rewrite $\frac{x^{2} x - 4 \cdot 4}{4 x}$ in a factored form.

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

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

$\frac{x^{2}}{4} \cdot \frac{1}{\frac{x^{2} x^{1} - 4 \cdot 4}{4 x}}$

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

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

Add $2$ and $1$ .

$\frac{x^{2}}{4} \cdot \frac{1}{\frac{x^{3} - 4 \cdot 4}{4 x}}$

Multiply $- 4$ by $4$ .

$\frac{x^{2}}{4} \cdot \frac{1}{\frac{x^{3} - 16}{4 x}}$

Step 3

Simplify terms.

Combine.

$\frac{x^{2} \cdot 1}{4 \frac{x^{3} - 16}{4 x}}$

Multiply $x^{2}$ by $1$ .

$\frac{x^{2}}{4 \frac{x^{3} - 16}{4 x}}$

Combine $4$ and $\frac{x^{3} - 16}{4 x}$ .

$\frac{x^{2}}{\frac{4 (x^{3} - 16)}{4 x}}$

Reduce the expression $\frac{4 (x^{3} - 16)}{4 x}$ by cancelling the common factors.

Cancel the common factor.

$\frac{x^{2}}{\frac{4 (x^{3} - 16)}{4 x}}$

Rewrite the expression.

$\frac{x^{2}}{\frac{x^{3} - 16}{x}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$x^{2} \frac{x}{x^{3} - 16}$

Step 5

Multiply $x^{2} \frac{x}{x^{3} - 16}$ .

Combine $x^{2}$ and $\frac{x}{x^{3} - 16}$ .

$\frac{x^{2} x}{x^{3} - 16}$

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

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

$\frac{x^{2} x^{1}}{x^{3} - 16}$

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

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

Add $2$ and $1$ .

$\frac{x^{3}}{x^{3} - 16}$

Simplify ((x^2)/4)/((x^2)/4-4/x)

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