Which is equivalent to left-parenthesis 5 StartRoot 2 EndRoot right-parenthesis left-parenthesis 3 StartRoot 6 EndRoot right-parenthesis?

Question

A.
8 StartRoot 8 EndRoot

B.
8 StartRoot 12 EndRoot

C.
15 StartRoot 8 EndRoot

D.
30 StartRoot 3 EndRoot

E.
60 StartRoot 3 EndRoot

Answer 1

To simplify the expression \( (5\sqrt{2})(3\sqrt{6}) \), we can use the properties of multiplication.

1. First, multiply the constants (5 and 3): \[ 5 \times 3 = 15 \]

2. Next, multiply the square roots: \[ \sqrt{2} \times \sqrt{6} = \sqrt{2 \times 6} = \sqrt{12} \]

3. Now combine the results: \[ (5\sqrt{2})(3\sqrt{6}) = 15\sqrt{12} \]

Now, we can simplify \(\sqrt{12}\): \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]

So, substituting this back, we have: \[ 15\sqrt{12} = 15 \times 2\sqrt{3} = 30\sqrt{3} \]

Therefore, the equivalent expression of \( (5\sqrt{2})(3\sqrt{6}) \) is: \[ 30\sqrt{3} \]

Thus, the correct answer is: D. \( 30\sqrt{3} \).