The problem asks to multiply and simplify the expression $8\sqrt{a}(6\sqrt{b} - 3)$. We are to assume that all variables represent nonnegative numbers.

1. Problem Description

The problem asks to multiply and simplify the expression $8\sqrt{a}(6\sqrt{b} - 3)$. We are to assume that all variables represent nonnegative numbers.

2. Solution Steps

We distribute $8\sqrt{a}$ to both terms inside the parentheses:

$$8\sqrt{a}(6\sqrt{b} - 3) = 8\sqrt{a} \cdot 6\sqrt{b} - 8\sqrt{a} \cdot 3$$

Now, we can multiply the coefficients and the radicals:

$$8\sqrt{a} \cdot 6\sqrt{b} = (8 \cdot 6) (\sqrt{a} \cdot \sqrt{b}) = 48\sqrt{ab}$$

and

$$8\sqrt{a} \cdot 3 = (8 \cdot 3)\sqrt{a} = 24\sqrt{a}$$

Therefore,

$$8\sqrt{a}(6\sqrt{b} - 3) = 48\sqrt{ab} - 24\sqrt{a}$$

3. Final Answer

$48\sqrt{ab} - 24\sqrt{a}$