# Simplify the following. y = \frac{x}{4 - \sqrt{16 + x})}

## Question:

Simplify the following.

{eq}\displaystyle y = \frac{x}{4 - \sqrt{16 + x})} {/eq}

## Simplification:

We are given a rational function and we need to find out the simplified form. Rational function has a denominator and a numerator.

To solve this problem, we'll multiply the function by the conjugate and cancel out the common factor.

We are given:

{eq}\displaystyle y = \frac{x}{4 - \sqrt{16 + x}} {/eq}

Multiply by the conjugate:

{eq}= \displaystyle \frac{x(4 - \sqrt{16 + x})}{(4 - \sqrt{16 + x}) (4 + \sqrt{16 + x})} {/eq}

{eq}= \displaystyle \frac{x(4 + \sqrt{16 + x})}{16 - (16+x)} {/eq}

{eq}= \displaystyle - \frac{x(4 + \sqrt{16 + x})}{x} {/eq}

Cancel out the common factor:

{eq}= \displaystyle -4 -\sqrt{16 + x} {/eq}

Therefore, the simplified form of the given function is {eq}\displaystyle y = -4 - \sqrt{16 + x}. {/eq}