Question

use the calculator below to approximate \\(\sqrt{84}\\) as a decimal to the hundredths place. note that you must do the approximation without using a square root button. your answer must be within a hundredth of the actual value.

Explanation:

Step1: Identify nearest perfect squares

We know $9^2=81$ and $10^2=100$, so $\sqrt{84}$ is between 9 and 10.

Step2: Narrow down to tenths place

Test $9.1^2=82.81$, $9.2^2=84.64$. Since $82.81<84<84.64$, $\sqrt{84}$ is between 9.1 and 9.2.

Step3: Refine to hundredths place

Calculate $9.16^2=9.16\times9.16=83.9056$
Calculate $9.17^2=9.17\times9.17=84.0889$
84 is closer to $9.16^2$ (difference: $84-83.9056=0.0944$) than to $9.17^2$ (difference: $84.0889-84=0.0889$; the two differences are very close, but 84 is between them, and we round to the hundredths place where the value is within 0.01 of the actual $\sqrt{84}$).

Answer:

9.17