What is the product of 3 and \( 10 \sqrt{40} \) in simplest radical form?

To find the product of \( 3 \) and \( 10 \sqrt{40} \), we begin by simplifying \( \sqrt{40} \). First, factor \( 40 \): \[ 40 = 4 \times 10 = 4 \times 2 \times 5 \] Now, we can simplify \( \sqrt{40} \): \[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10} \] Next, substitute \( \sqrt{40} \) back into the expression: \[ 10 \sqrt{40} = 10 \times 2 \sqrt{10} = 20 \sqrt{10} \] Now we multiply this by \( 3 \): \[ 3 \times (20 \sqrt{10}) = 60 \sqrt{10} \] Thus, the product of \( 3 \) and \( 10 \sqrt{40} \) in simplest radical form is: \[ \boxed{60 \sqrt{10}} \]