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Which expression will equal a rational product even though it is multiplying an irrational number by another irrational number?

A. [tex]\sqrt{11} \times \sqrt{11}[/tex]
B. [tex]4.7813265 \ldots \times \sqrt{5}[/tex]
C. [tex]\pi \times 3.785492 \ldots[/tex]
D. [tex]\sqrt{21} \times \pi[/tex]


Sagot :

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.