Question

Without calculating determine whether each
product is rational or irrational
2. 3̅ 𝑥 5. 7̅
14 x 0.010203

Answer 1

To determine whether the product of two numbers is rational or irrational, we can analyze each number individually.

1. 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.

2. 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:
1. \(3.\overline{3} \times 5.\overline{7}\) is rational.
2. \(14 \times 0.010203\) is rational.