Question

which of the following are irrational numbers? -√4, √17, -38, 10

Explanation:

Step1: Recall irrational number definition

An irrational number is a number that cannot be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq0$, and its decimal expansion is non - repeating and non - terminating. Rational numbers include integers, fractions, and perfect square roots.

Step2: Analyze $-\sqrt{4}$

Simplify $-\sqrt{4}$, since $4 = 2\times2$, then $\sqrt{4}=2$, so $-\sqrt{4}=- 2$, which is an integer. Integers are rational numbers.

Step3: Analyze $\sqrt{17}$

17 is not a perfect square (since $4^2 = 16$ and $5^2=25$, and there is no integer $n$ such that $n^2 = 17$). So $\sqrt{17}$ has a non - repeating, non - terminating decimal expansion, so $\sqrt{17}$ is an irrational number.

Step4: Analyze $- 38$

-38 is an integer. Integers are rational numbers because they can be written as $\frac{-38}{1}$.

Step5: Analyze $10$

10 is an integer. Integers are rational numbers because they can be written as $\frac{10}{1}$.

Answer:

The only irrational number among the given numbers is $\boldsymbol{\sqrt{17}}$. The correct selection should be only $\sqrt{17}$ (the other options $-\sqrt{4}$, $- 38$, and $10$ are rational numbers).