Let's examine each expression in detail to determine which one results in a rational number when multiplying two irrational numbers together.
### Expression 1: \(\sqrt{11} \times \sqrt{11}\)
The product \(\sqrt{11} \times \sqrt{11}\) can be simplified using the properties of square roots:
[tex]\[ \sqrt{11} \times \sqrt{11} = (\sqrt{11})^2 = 11 \][/tex]
This results in the rational number 11.
### Expression 2: \(4.7813265 \ldots \times \sqrt{5}\)
Next, let's look at the expression involving \(4.7813265 \ldots\) (a decimal approximation) and \(\sqrt{5}\):
[tex]\[ 4.7813265 \ldots \times \sqrt{5} \][/tex]
Upon computing this product, we obtain approximately:
[tex]\[ 10.691371076621147 \][/tex]
Since this is not a whole number and does not have a repeating decimal pattern, it is an irrational number.
### Expression 3: \(\pi \times 3.785492 \ldots\)
Now, consider the product involving \(\pi\) and another decimal approximation:
[tex]\[ \pi \times 3.785492 \ldots \][/tex]
With the calculation, we get approximately:
[tex]\[ 11.892473857422933 \][/tex]
Again, this does not result in a whole number or a repeating decimal, indicating it is irrational.
### Expression 4: \(\sqrt{21} \times \pi\)
Finally, we multiply \(\sqrt{21}\) with \(\pi\):
[tex]\[ \sqrt{21} \times \pi \][/tex]
The computation gives approximately:
[tex]\[ 14.396586137792408 \][/tex]
This too is an irrational number as it does not simplify to a rational form.
### Conclusion
Among all the expressions, only the first expression:
[tex]\[ \sqrt{11} \times \sqrt{11} = 11 \][/tex]
results in a rational number, which is [tex]\(11\)[/tex]. The rest of the expressions yield irrational numbers. Hence, [tex]\(\sqrt{11} \times \sqrt{11}\)[/tex] is the only one where the product of two irrational numbers results in a rational number.
