To find the area of the garden, we need to multiply the two dimensions together:
\[ \text{Area} = \sqrt{300} \times \sqrt{75} \]
Using the property of square roots that \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \), we can combine the square roots:
\[ \text{Area} = \sqrt{300 \times 75} \]
Now, we can calculate the product inside the square root:
\[ 300 \times 75 = 22500 \]
Next, we need to find the square root of \( 22500 \). We can simplify this step by first factorizing the number:
\[ 22500 = 225 \times 100 \]
Now we can take the square root:
\[ \sqrt{22500} = \sqrt{225 \times 100} = \sqrt{225} \times \sqrt{100} = 15 \times 10 = 150 \]
Thus, the area of the garden in simplest radical form is:
\[ \boxed{150} \]