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 64 is a perfect square, $sqrt{64}$ is irrational.
since it is an integer, - 1 is rational.
since it is a repeating decimal, $5.overline{34}$ is irrational.
none of the above statements are true.

Explanation:

Step1: Recall the definition of rational and irrational numbers

A rational number is a number that can be written as a ratio of two integers $\frac{p}{q}$ ($q
eq0$). Integers are rational numbers since they can be written as $\frac{n}{1}$ where $n$ is an integer. A non - perfect - square root is irrational, and a perfect - square root is rational. Repeating decimals are rational.

Step2: Analyze each statement

• $\sqrt{14}$: Since 14 is not a perfect square, $\sqrt{14}$ is irrational, so the first statement is false.
• $\frac{3}{4}$: By the definition of a rational number (a ratio of two integers), $\frac{3}{4}$ is rational, so the second statement is true.
• $\sqrt{64}=8$, and 8 is an integer and thus rational. So the third statement is false.
• - 1 is an integer. An integer can be written as $\frac{-1}{1}$, so - 1 is rational. The fourth statement is true.
• $5.\overline{34}$ is a repeating decimal. Repeating decimals can be written as a ratio of two integers, so it is rational. The fifth statement is false.

Answer:

• Since it is a ratio of two integers, $\frac{3}{4}$ is rational.
• Since it is an integer, - 1 is rational.