Question

select the two correct answers.
a number line is shown.

which expressions have a value between 2 and 3 on the number line?

□ √5
□ √3
□ √8
□ π
□ √2

Explanation:

Step1: Analyze \(\sqrt{5}\)

We know that \(2^2 = 4\) and \(3^2=9\). For \(\sqrt{5}\), since \(4<5 < 9\), taking square roots (and since square root is an increasing function for non - negative numbers), we have \(2=\sqrt{4}<\sqrt{5}<\sqrt{9} = 3\). So \(\sqrt{5}\) is between 2 and 3.

Step2: Analyze \(\sqrt{3}\)

We know that \(1^2 = 1\) and \(2^2=4\). Since \(1<3 < 4\), then \(\sqrt{1}=1<\sqrt{3}<\sqrt{4}=2\). So \(\sqrt{3}\) is between 1 and 2, not between 2 and 3.

Step3: Analyze \(\sqrt{8}\)

We know that \(2^2 = 4\) and \(3^2 = 9\). Since \(4<8<9\), then \(\sqrt{4}=2<\sqrt{8}<\sqrt{9}=3\). So \(\sqrt{8}\) is between 2 and 3.

Step4: Analyze \(\pi\)

We know that \(\pi\approx3.14\), which is greater than 3, so it is not between 2 and 3.

Step5: Analyze \(\sqrt{2}\)

We know that \(1^2 = 1\) and \(2^2=4\). Since \(1<2<4\), then \(\sqrt{1}=1<\sqrt{2}<\sqrt{4}=2\). So \(\sqrt{2}\) is between 1 and 2, not between 2 and 3.

Answer:

\(\sqrt{5}\), \(\sqrt{8}\)