Find the value of the following expression:

Let us solve the nested radical expression in the numerator step-by-step from the innermost root:
Step 1: Simplify $\sqrt{225}$

$$\sqrt{225}=15$$

Step 2: Simplify $\sqrt{154+15}$

$$\sqrt{169}=13$$

Step 3: Simplify $\sqrt{108+13}$

$$\sqrt{121}=11$$

Step 4: Simplify $\sqrt{25+11}$

$$\sqrt{36}=6$$

Step 5: Simplify $\sqrt{10+6}$

$$\sqrt{16}=4$$

So, the numerator evaluates to $4$.

Step 6: Evaluate the denominator $\sqrt[3]{8}$

$$\sqrt[3]{8}=2$$

Step 7: Substitute these back into the original expression:

$$\text{Value}=8\times \frac{4}{2}=8\times 2=16$$