To determine whether the product of two numbers is rational or irrational, we can analyze each number individually.
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For the first product: \(3.\overline{3} \times 5.\overline{7}\)
- \(3.\overline{3}\) (which is equivalent to \(3.3333...\)) is a repeating decimal. Repeating decimals can be expressed as fractions, which means they are rational numbers.
- \(5.\overline{7}\) (which is equivalent to \(5.7777...\)) is also a repeating decimal, and thus is also a rational number.
The product of two rational numbers is always rational. Therefore, \(3.\overline{3} \times 5.\overline{7}\) is rational.
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For the second product: \(14 \times 0.010203\)
- \(14\) is an integer, which is also a rational number.
- \(0.010203\) is a finite decimal, which can be expressed as a fraction (specifically, \( \frac{10203}{1000000} \)), and thus it is also a rational number.
Again, the product of two rational numbers is always rational. Therefore, \(14 \times 0.010203\) is also rational.
In summary:
- \(3.\overline{3} \times 5.\overline{7}\) is rational.
- \(14 \times 0.010203\) is rational.