Algebra Examples

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

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.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Reorder the factors of $x \cdot 4$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

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

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Step 2.5.1

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

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Step 2.5.1.1

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

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Step 2.5.1.1.1

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

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

Step 2.5.1.1.2

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

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

Step 2.5.1.2

Add $2$ and $1$ .

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

Step 2.5.2

Multiply $- 4$ by $4$ .

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

Step 3

Simplify terms.

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Step 3.1

Combine.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

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

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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Step 5.2.1.1

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

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

Step 5.2.1.2

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

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

Step 5.2.2

Add $2$ and $1$ .

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