Question

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

show your work

Answer 1

To determine whether the products of the given numbers are rational or irrational, we need to analyze the numbers involved.

1. **Identifying the Numbers:**
- The notation \(3̅\) and \(7̅\) indicates repeating decimals:
- \(3̅ = 3.333...\) which is the same as \(\frac{10}{3}\).
- \(7̅ = 7.777...\) which is the same as \(\frac{70}{9}\).
- The other numbers mentioned are:
- \(5\), which is a rational number since it can also be represented as \(\frac{5}{1}\).
- \(14\), which is also a rational number (i.e., \(\frac{14}{1}\)).
- \(0.010203\), which is a non-repeating decimal but it can be expressed as a fraction (e.g., \(\frac{10203}{1000000}\)), so it is also rational.

2. **Analyzing the Products:**
- **First Product:** \(3̅ \times 5 \times 7̅\)
- Let's write it in terms of fractions:
- \(3̅ = \frac{10}{3}\)
- \(7̅ = \frac{70}{9}\)
- Thus the product can be expressed as:
\[
3̅ \times 5 \times 7̅ = \left(\frac{10}{3}\right) \times 5 \times \left(\frac{70}{9}\right)
\]
- This results in a product of fractions, which is always rational.

- **Second Product:** \(14 \times 0.010203\)
- We already established both \(14\) (rational) and \(0.010203\) (rational):
- Since both are rational numbers, the product of two rational numbers is also rational.
\[
14 \times 0.010203
\]

3. **Conclusion:**
- **First Product:** The product \(3̅ \times 5 \times 7̅\) is rational.
- **Second Product:** The product \(14 \times 0.010203\) is rational.

Therefore, both products are rational.