Question

without using a calculator, determine which expression has the value shown by the point on the number line below. the options are \\(sqrt{92}\\), \\(sqrt{55}\\), \\(sqrt{44}\\), \\(sqrt{77}\\).

Explanation:

Step1: Locate point on number line

The point is between 9 and 10, so $9 < x < 10$.

Step2: Square all inequality terms

$$9^2 < x^2 < 10^2$$
$$81 < x^2 < 100$$

Step3: Match to square root values

We need $\sqrt{n}$ where $81 < n < 100$. Check options:

• $\sqrt{92}$: $92$ is between 81 and 100
• $\sqrt{55}$: $55 < 81$, too small
• $\sqrt{44}$: $44 < 81$, too small
• $\sqrt{77}$: $77 < 81$, too small

Answer:

$\sqrt{92}$