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:

Step1: Define rational/irrational numbers

Rational numbers can be written as $\frac{p}{q}$ where $p,q$ are integers, $q
eq0$; they include integers, fractions, terminating/repeating decimals. Irrational numbers cannot be written this way, e.g., square roots of non-perfect squares.

Step2: Evaluate statement 1

$\sqrt{14}$: 14 is not a perfect square, so $\sqrt{14}$ is irrational. Statement is false.

Step3: Evaluate statement 2

$\frac{3}{4}$: Ratio of integers 3 and 4, fits rational definition. Statement is true.

Step4: Evaluate statement 3

$\sqrt{9}=3$, an integer (rational). Statement is false.

Step5: Evaluate statement 4

$-6$: Integers are rational ($\frac{-6}{1}$). Statement is true.

Step6: Evaluate statement 5

$7.\overline{42}$: Repeating decimals are rational. Statement is false.

Step7: Evaluate final option

True statements exist, so this is false.

Answer:

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