What is the product of the following in simplest radical 4 8 radical 5 times 3 radical 32

Answer 1

To simplify the expression \( 4 \cdot 8 \sqrt{5} \cdot 3 \sqrt{32} \), we can follow these steps:

1. Multiply the coefficients: \[ 4 \cdot 8 \cdot 3 = 96 \]

2. Simplify the radicals:

   • Start with \( \sqrt{32} \): \[ \sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} \]

3. Substituting \( \sqrt{32} \) into the expression: \[ 4 \cdot 8 \sqrt{5} \cdot 3 \cdot 4\sqrt{2} = 4 \cdot 8 \cdot 3 \cdot 4 \cdot \sqrt{5} \cdot \sqrt{2} \]

4. Multiply the coefficients again: \[ 4 \cdot 8 \cdot 3 \cdot 4 = 384 \]

5. Combine the radicals: \[ \sqrt{5} \cdot \sqrt{2} = \sqrt{10} \]

6. Combine the final result: \[ 384\sqrt{10} \]

Thus, the product in its simplest radical form is: \[ \boxed{384\sqrt{10}} \]