[tex]f(x)=\sqrt[]{6x+16}[/tex]
First, we need to solve for f(x + h). To do this, we only have to replace x with x + h, like this.
[tex]\begin{gathered} f(x+h)=\sqrt[]{6(x+h)+16} \\ f(x+h)=\sqrt[]{6x+6h+16} \end{gathered}[/tex]
Now, we have a value for f(x) and f(x+h). Let's substitute this data to the equation being asked.
[tex]\frac{f(x+h)-f(x)}{h}=\frac{\sqrt[]{6x+6h+16}-\sqrt[]{6x+16}}{h}[/tex]
To simplify:
[tex]\begin{gathered} \frac{f(x+h)-f(x)}{h}=\frac{\sqrt[]{6x+6h+16}-\sqrt[]{6x+16}}{h} \\ \frac{f(x+h)-f(x)}{h}=\frac{\sqrt[]{2(3x+3h+8)}-\sqrt[]{2(3x+8)}}{h} \\ \frac{f(x+h)-f(x)}{h}=\frac{\sqrt[]{2}(\sqrt[]{3x+3h+8})-\sqrt[]{2}(\sqrt[]{3x+8})}{h} \\ \frac{f(x+h)-f(x)}{h}=\frac{\sqrt[]{2}(\sqrt[]{3x+3h+8}-\sqrt[]{3x+8})}{h} \end{gathered}[/tex]
The simplified equation is the last equation above.
