Let's solve the given problem step-by-step.
We are given the expression:
[tex]\[ \sqrt{3} \cdot \sqrt{5} \cdot \sqrt{6} \][/tex]
First, let's recognize a property of square roots:
[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]
Using this property, we can combine the square roots. Specifically:
[tex]\[ \sqrt{3} \cdot \sqrt{5} \cdot \sqrt{6} = \sqrt{3 \cdot 5 \cdot 6} \][/tex]
Next, we calculate the product inside the square root:
[tex]\[ 3 \cdot 5 = 15 \][/tex]
[tex]\[ 15 \cdot 6 = 90 \][/tex]
So, the expression simplifies to:
[tex]\[ \sqrt{90} \][/tex]
Now, we need to determine which of the given choices is equivalent to [tex]\( \sqrt{90} \)[/tex].
Let's consider each option:
A. [tex]\( 9 \sqrt{10} \)[/tex]
To compare, we can consider the numerical simplification of [tex]\( 9 \sqrt{10} \)[/tex]:
[tex]\[ 9 \sqrt{10} = 9 \cdot \sqrt{10} \approx 28.4605 \][/tex]
This does not match [tex]\( \sqrt{90} \)[/tex].
B. [tex]\( 3 \sqrt{5} \)[/tex]
Comparing numerically:
[tex]\[ 3 \sqrt{5} = 3 \cdot \sqrt{5} \approx 6.7082 \][/tex]
This does not match [tex]\( \sqrt{90} \)[/tex].
C. [tex]\( 10 \sqrt{3} \)[/tex]
Comparing numerically:
[tex]\[ 10 \sqrt{3} = 10 \cdot \sqrt{3} \approx 17.3205 \][/tex]
This does not match [tex]\( \sqrt{90} \)[/tex].
D. [tex]\( 3 \sqrt{10} \)[/tex]
Comparing numerically:
[tex]\[ 3 \sqrt{10} = 3 \cdot \sqrt{10} \approx 9.4868 \][/tex]
This matches the value of [tex]\( \sqrt{90} \)[/tex], since:
[tex]\[ \sqrt{90} \approx 9.4868 \][/tex]
Therefore, the choice that is equivalent to the product [tex]\(\sqrt{3} \cdot \sqrt{5} \cdot \sqrt{6}\)[/tex] is:
[tex]\[ \boxed{3 \sqrt{10}} \][/tex]
