To find the product of \( 7 \sqrt{10} \) and \( 10 \sqrt{15} \), we can multiply the two expressions together:
\[ (7 \sqrt{10}) \times (10 \sqrt{15}) \]
First, multiply the coefficients (numerical parts):
\[ 7 \times 10 = 70 \]
Next, multiply the square roots:
\[ \sqrt{10} \times \sqrt{15} = \sqrt{10 \times 15} = \sqrt{150} \]
Now combine both parts:
\[ 70 \sqrt{150} \]
Next, let's simplify \( \sqrt{150} \):
\[ \sqrt{150} = \sqrt{25 \times 6} = \sqrt{25} \times \sqrt{6} = 5 \sqrt{6} \]
Now, replace \( \sqrt{150} \) in the product:
\[ 70 \sqrt{150} = 70 \times 5 \sqrt{6} = 350 \sqrt{6} \]
Thus, the product of \( 7 \sqrt{10} \) and \( 10 \sqrt{15} \) in simplest radical form is:
\[ \boxed{350 \sqrt{6}} \]