Question

between which two consecutive whole numbers does \\(\sqrt{40}\\) lie? fill out the sentence below to justify your answer and use your mouse to drag \\(\sqrt{40}\\) to an approximately correct location on the number line.

answer
since \\(\sqrt{\square} = \square\\) and \\(\sqrt{\square} = \square\\) it is known that \\(\sqrt{40}\\) is between \\(\square\\) and \\(\square\\).

Explanation:

Step1: Find the perfect square less than 40

We know that \( 6^2 = 36 \), so \( \sqrt{36} = 6 \).

Step2: Find the perfect square greater than 40

We know that \( 7^2 = 49 \), so \( \sqrt{49} = 7 \).

Step3: Determine the range of \( \sqrt{40} \)

Since \( 36 < 40 < 49 \), taking square roots (which is a monotonically increasing function for non - negative numbers), we have \( \sqrt{36}<\sqrt{40}<\sqrt{49} \), that is \( 6 < \sqrt{40}<7 \).

Answer:

Since \(\sqrt{\boldsymbol{36}}=\boldsymbol{6}\) and \(\sqrt{\boldsymbol{49}}=\boldsymbol{7}\) it is known that \(\sqrt{40}\) is between \(\boldsymbol{6}\) and \(\boldsymbol{7}\).