To determine whether the products of the given numbers are rational or irrational, we need to analyze the numbers involved.
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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.
- The notation \(3̅\) and \(7̅\) indicates repeating decimals:
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Analyzing the Products:
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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.
- Let's write it in terms of fractions:
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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. \[ 14 \times 0.010203 \]
- We already established both \(14\) (rational) and \(0.010203\) (rational):
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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.
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