To determine which expression is equivalent to [tex]\(5 \sqrt{6} \cdot 2 \sqrt{3}\)[/tex], let's carefully walk through each step of the simplification process.
1. Combine the constants:
We have the constants 5 and 2 in the given expression [tex]\(5 \sqrt{6} \cdot 2 \sqrt{3}\)[/tex].
Combining these gives:
[tex]\[
5 \times 2 = 10
\][/tex]
2. Combine the square root terms:
Next, we combine the square root terms [tex]\(\sqrt{6}\)[/tex] and [tex]\(\sqrt{3}\)[/tex]. Using the property of square roots [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex], we get:
[tex]\[
\sqrt{6} \cdot \sqrt{3} = \sqrt{6 \cdot 3} = \sqrt{18}
\][/tex]
3. Simplify [tex]\(\sqrt{18}\)[/tex]:
The square root of 18 can be further simplified. We recognize that:
[tex]\[
\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3 \cdot \sqrt{2}
\][/tex]
4. Combine the constants with the simplified square root:
We had previously combined the constants to get 10. Now we multiply this by the simplified square root term we just found:
[tex]\[
10 \cdot (3 \sqrt{2}) = 10 \times 3 \sqrt{2} = 30 \sqrt{2}
\][/tex]
Thus, the original expression [tex]\(5 \sqrt{6} \cdot 2 \sqrt{3}\)[/tex] simplifies to:
[tex]\[
30 \sqrt{2}
\][/tex]
Therefore, the correct answer is:
[tex]\[
30 \sqrt{2}
\][/tex]
