Use the product rule to simplify the radical. \( \begin{array}{l}3 \sqrt{45} \\ 3 \sqrt{45}=\square \\ \text { (Simplify your answer. Type an exact answer, using radicals as needed.) }\end{array} \)

To simplify \( 3 \sqrt{45} \), we can start by recognizing that \( 45 \) can be factored into \( 9 \times 5 \). Since \( 9 \) is a perfect square, we have: \[ 3 \sqrt{45} = 3 \sqrt{9 \times 5} = 3 \sqrt{9} \sqrt{5} = 3 \times 3 \sqrt{5} = 9 \sqrt{5}. \] So, \( 3 \sqrt{45} = 9 \sqrt{5} \). To further enhance our learning, let’s explore the foundational aspects. The product rule of square roots states that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This is particularly useful when simplifying expressions involving square roots, as it helps break down complex radicals into simpler parts. There’s also a common mistake in this process! Students might forget to simplify completely or overlook the perfect squares hidden within the radical's components. Always remember to look for factors of your radicand that are perfect squares for the easiest simplification!