Evaluate \(\sqrt{15} \times (\sqrt{3})^3\)

To evaluate this expression, \(\sqrt{15} \times (\sqrt{3})^3\), we'll first simplify \((\sqrt{3})^3\) and then multiply the results.

Recall that \((\sqrt{a})^b = a^{b/2}\). In this case, \(a = 3\) and \(b = 3\), so we have:

\[(\sqrt{3})^3 = 3^{3/2}\]

Now, we can rewrite the exponent as a product of two fractions:

\[3^{3/2} = 3^{1 \times (3/2)} = (3^1)^{3/2}\]

We know that \(3^1 = 3\), so the expression simplifies to:

\[(3^1)^{3/2} = 3^{3/2}\]

Next, let's multiply \(\sqrt{15}\) by \(3^{3/2}\):

\[\sqrt{15} \times 3^{3/2}\]

Since we're multiplying two square roots, we can combine them using the product property of square roots, which states that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\). In this case, \(a = 15\) and \(b = 3^{3/2}\), so we have:

\[\sqrt{15} \times 3^{3/2} = \sqrt{15 \times 3^{3/2}}\]

The expression inside the square root can be simplified as:

\[15 \times 3^{3/2} = 15 \times 3 \times \sqrt{3} = 45 \times \sqrt{3}\]

Finally, we have:

\[\sqrt{15} \times (\sqrt{3})^3 = \sqrt{45 \times \sqrt{3}}\]

Option C, \(9\sqrt{5}\), is the correct answer. To see this, we can rewrite the expression as follows:

\[\sqrt{45 \times \sqrt{3}} = \sqrt{9 \times 5 \times \sqrt{3}} = \sqrt{(9 \times \sqrt{3}) \times 5} = 9\sqrt{5}\]