To determine which expression represents a rational number, let's evaluate each of the given expressions step-by-step.
### Expression A: [tex]\((2 \sqrt{3})(3 \sqrt{2})\)[/tex]
[tex]\[
(2 \sqrt{3})(3 \sqrt{2}) = 2 \cdot 3 \cdot \sqrt{3} \cdot \sqrt{2} = 6 \sqrt{6}
\][/tex]
The term [tex]\( \sqrt{6} \)[/tex] is irrational, therefore [tex]\( 6\sqrt{6} \)[/tex] is irrational.
### Expression B: [tex]\((2 \sqrt{6})(3 \sqrt{2})\)[/tex]
[tex]\[
(2 \sqrt{6})(3 \sqrt{2}) = 2 \cdot 3 \cdot \sqrt{6} \cdot \sqrt{2} = 6 \sqrt{12}
\][/tex]
Since [tex]\( \sqrt{12} = \sqrt{4 \times 3} = 2 \sqrt{3} \)[/tex], and [tex]\( \sqrt{3} \)[/tex] is irrational, [tex]\( \sqrt{12} \)[/tex] is also irrational. Hence, [tex]\( 6 \sqrt{12} \)[/tex] is irrational.
### Expression C: [tex]\((6 \sqrt{3})(6 \sqrt{2})\)[/tex]
[tex]\[
(6 \sqrt{3})(6 \sqrt{2}) = 6 \cdot 6 \cdot \sqrt{3} \cdot \sqrt{2} = 36 \sqrt{6}
\][/tex]
Again, [tex]\( \sqrt{6} \)[/tex] is irrational, making [tex]\( 36 \sqrt{6} \)[/tex] irrational.
### Expression D: [tex]\((2 \sqrt{3})(2 \sqrt{3})\)[/tex]
[tex]\[
(2 \sqrt{3})(2 \sqrt{3}) = 2 \cdot 2 \cdot \sqrt{3} \cdot \sqrt{3} = 4 \cdot 3 = 12
\][/tex]
Here, [tex]\( 12 \)[/tex] is a rational number.
### Expression E: [tex]\((3 \sqrt{3})(6 \sqrt{6})\)[/tex]
[tex]\[
(3 \sqrt{3})(6 \sqrt{6}) = 3 \cdot 6 \cdot \sqrt{3} \cdot \sqrt{6} = 18 \sqrt{18}
\][/tex]
Since [tex]\( \sqrt{18} = \sqrt{9 \times 2} = 3 \sqrt{2} \)[/tex] and [tex]\( \sqrt{2} \)[/tex] is irrational, [tex]\( \sqrt{18} \)[/tex] is also irrational. Thus, [tex]\( 18 \sqrt{18} \)[/tex] is irrational.
Among all the expressions, only Expression D results in a rational number. Therefore, the expression that represents a rational number is:
[tex]\[
(2 \sqrt{3})(2 \sqrt{3}) = 12
\][/tex]
So, the correct answer is Expression D.
