Question

check all statements that are true.

since 14 is not a perfect square, $sqrt{14}$ is rational.

since it is a ratio of two integers, $\frac{3}{4}$ is rational.

since 9 is a perfect square, $sqrt{9}$ is irrational.

since it is an integer, $-6$ is rational.

since it is a repeating decimal, $7.overline{42}$ is irrational.

none of the above statements are true.

Explanation:

1. A non-perfect square's square root is irrational, so $\sqrt{14}$ is irrational, making the first statement false.
2. Rational numbers are defined as ratios of two integers (denominator non-zero), so $\frac{3}{4}$ fits this, making the second statement true.
3. $\sqrt{9}=3$, an integer and rational number, so the third statement is false.
4. All integers can be written as a ratio of themselves and 1, so $-6$ is rational, making the fourth statement true.
5. Repeating decimals are rational numbers, so $7.\overline{42}$ is rational, making the fifth statement false.
6. Since some statements are true, the last statement is false.

Answer:

• Since it is a ratio of two integers, $\frac{3}{4}$ is rational.
• Since it is an integer, $-6$ is rational.