Find the domain of the expression. \[ \begin{array}{l}\frac{\sqrt{x-4}}{x+6} \\ x \neq 4 \\ x>-6 \\ x \neq-6 \\ \text { all real numbers } \\ x \geq 4\end{array} \]

To find the domain of the expression \(\frac{\sqrt{x-4}}{x+6}\), we need to identify the restrictions placed on \(x\) by both the numerator and the denominator. First, for the square root in the numerator \(\sqrt{x-4}\), the expression under the square root must be non-negative. This gives us the condition: \[ x - 4 \geq 0 \implies x \geq 4. \] Next, we look at the denominator \(x + 6\). For the expression to be defined, the denominator cannot be equal to zero, which leads to: \[ x + 6 \neq 0 \implies x \neq -6. \] Combining these conditions, we see that \(x\) must be at least 4, and since \(x\) cannot be \(-6\), we don't have to worry about that since it’s already below the threshold of 4. So, the domain of the expression is: \[ \text{Domain: } [4, \infty) \text{ (all real numbers } x \text{ such that } x \geq 4). \]