The problem is to multiply and simplify the expression $8\sqrt{a}(2\sqrt{b} - 6)$, assuming that all variables represent nonnegative numbers.

1. Problem Description

The problem is to multiply and simplify the expression $8\sqrt{a}(2\sqrt{b} - 6)$, assuming that all variables represent nonnegative numbers.

2. Solution Steps

We need to distribute the term $8\sqrt{a}$ to both terms inside the parenthesis:

$8\sqrt{a}(2\sqrt{b} - 6) = 8\sqrt{a} \cdot 2\sqrt{b} - 8\sqrt{a} \cdot 6$

Then, we multiply the coefficients:

$8 \cdot 2 \cdot \sqrt{a} \cdot \sqrt{b} - 8 \cdot 6 \cdot \sqrt{a}$

$16\sqrt{a}\sqrt{b} - 48\sqrt{a}$

Using the property $\sqrt{a}\sqrt{b} = \sqrt{ab}$:

$16\sqrt{ab} - 48\sqrt{a}$

3. Final Answer

$16\sqrt{ab} - 48\sqrt{a}$