Question

simplify.
$2\sqrt{3} \cdot 3\sqrt{15}$

Explanation:

Step1: Multiply the coefficients and the radicals separately

First, multiply the coefficients (the numbers outside the square roots): \(2\times3 = 6\). Then, multiply the radicals (the square root terms): \(\sqrt{3}\times\sqrt{15}\). Using the property of square roots \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\), we get \(\sqrt{3\times15}=\sqrt{45}\). So now we have \(6\sqrt{45}\).

Step2: Simplify the square root

Simplify \(\sqrt{45}\). We can factor 45 into \(9\times5\), and since \(\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}\) (using the same square root property), and \(\sqrt{9} = 3\), so \(\sqrt{45}=3\sqrt{5}\).

Step3: Multiply the coefficient with the simplified radical

Now, multiply the coefficient 6 with the 3 from \(\sqrt{45}=3\sqrt{5}\). So \(6\times3\sqrt{5}=18\sqrt{5}\).

Answer:

\(18\sqrt{5}\)