Simplify (1+4/x)/(1-16/(x^2))

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

Step 1

Multiply the numerator and denominator of the complex fraction by $x^{2}$ .

Multiply $\frac{1 + \frac{4}{x}}{1 - \frac{16}{x^{2}}}$ by $\frac{x^{2}}{x^{2}}$ .

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

Factor $x$ out of $x^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x^{2}$ .

Move the leading negative in $- \frac{16}{x^{2}}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

Factor $x$ out of $x^{2} \cdot 1 + x \cdot 4$ .

Factor $x$ out of $x^{2} \cdot 1$ .

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

Factor $x$ out of $x \cdot 4$ .

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

Factor $x$ out of $x (x \cdot 1) + x (4)$ .

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

Multiply $x$ by $1$ .

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

Step 5

Simplify the denominator.

Rewrite $x^{2} \cdot 1$ as ${(x \cdot 1)}^{2}$ .

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

Rewrite $16$ as $4^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x \cdot 1$ and $b = 4$ .

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

Simplify.

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

$\frac{x (x + 4)}{(x + 4) (x - 4)}$

Step 6

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{x (x + 4)}{(x + 4) (x - 4)}$

Rewrite the expression.

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

Simplify (1+4/x)/(1-16/(x^2))

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