Question

check all statements that are true.
□since 16 is a perfect square, \\(\sqrt{16}\\) is rational.
□since it is a repeating decimal, \\(15.\overline{2}\\) is irrational.
□since 14 is not a perfect square, \\(\sqrt{14}\\) is irrational.
□since it is a ratio of two integers, \\(\frac{3}{7}\\) is irrational.
□since it is a terminating decimal, 8.31 is rational.
□none of the above statements are true.

Explanation:

1. For "Since 16 is a perfect square, $\sqrt{16}$ is rational": $\sqrt{16} = 4$, and 4 is a rational number (can be expressed as $\frac{4}{1}$), so this statement is true.
2. For "Since it is a repeating decimal, $15.\overline{2}$ is irrational": Repeating decimals are rational (can be expressed as a fraction), so this statement is false.
3. For "Since 14 is not a perfect square, $\sqrt{14}$ is irrational": Non - perfect square square roots are irrational, so this statement is true.
4. For "Since it is a ratio of two integers, $\frac{3}{7}$ is irrational": A ratio of two integers is a rational number by definition, so this statement is false.
5. For "Since it is a terminating decimal, 8.31 is rational": Terminating decimals are rational (can be expressed as a fraction, e.g., $8.31=\frac{831}{100}$), so this statement is true.

Answer:

• Since 16 is a perfect square, $\sqrt{16}$ is rational.
• Since 14 is not a perfect square, $\sqrt{14}$ is irrational.
• Since it is a terminating decimal, 8.31 is rational.