To find the product of \( 8\sqrt{5} \) and \( 3\sqrt{32} \), we start by multiplying the coefficients and the radicals separately.
-
Multiply the coefficients: \[ 8 \times 3 = 24 \]
-
Multiply the radicals: \[ \sqrt{5} \times \sqrt{32} = \sqrt{5 \times 32} \]
-
Calculate \(5 \times 32\): \[ 5 \times 32 = 160 \] So, we have: \[ \sqrt{5} \times \sqrt{32} = \sqrt{160} \]
-
Now combine the results: \[ 8\sqrt{5} \times 3\sqrt{32} = 24\sqrt{160} \]
-
Simplify \(\sqrt{160}\): We first factor \(160\): \[ 160 = 16 \times 10 = 16 \times (2 \times 5) \] Since \(16\) is a perfect square, we can simplify: \[ \sqrt{160} = \sqrt{16 \times 10} = \sqrt{16} \times \sqrt{10} = 4\sqrt{10} \]
-
Substituting back: We now substitute \(\sqrt{160}\) back into our expression: \[ 24\sqrt{160} = 24 \times 4\sqrt{10} = 96\sqrt{10} \]
Thus, the product \(8\sqrt{5} \times 3\sqrt{32}\) in simplest radical form is: \[ \boxed{96\sqrt{10}} \]