Question

rationalize the denominator. assume that all variables represent positive real numbers.
\\(\frac{-3}{\sqrt{x} - 4}\\)

Explanation:

Step1: Identify the conjugate of the denominator

The denominator is $\sqrt{x} - 4$, so its conjugate is $\sqrt{x} + 4$. We multiply the numerator and the denominator by this conjugate to rationalize the denominator.
$$\frac{-3}{\sqrt{x} - 4} \times \frac{\sqrt{x} + 4}{\sqrt{x} + 4}$$

Step2: Multiply the numerators and the denominators

Multiply the numerators: $-3\times(\sqrt{x} + 4)=-3\sqrt{x}-12$
Multiply the denominators using the difference of squares formula $(a - b)(a + b)=a^2 - b^2$, where $a = \sqrt{x}$ and $b = 4$. So, $(\sqrt{x})^2-4^2=x - 16$
Putting it together, we get:
$$\frac{-3\sqrt{x}-12}{x - 16}$$

Answer:

$\frac{-3\sqrt{x}-12}{x - 16}$