Question

use the calculator below to approximate \\(\sqrt{134}\\) as a decimal to the tenths place. note that you must do the approximation without using a square root button. your answer must be within a tenth of the actual value. \\(\sqrt{134}\approx\square\\)

Explanation:

Step1: Find nearest perfect squares

We know that $11^2 = 121$ and $12^2 = 144$. Since $121 < 134 < 144$, $\sqrt{134}$ is between 11 and 12.

Step2: Test midpoint value

Calculate $11.5^2$:
$11.5^2 = (11 + 0.5)^2 = 11^2 + 2\times11\times0.5 + 0.5^2 = 121 + 11 + 0.25 = 133.25$

Step3: Test value above midpoint

Calculate $11.6^2$:
$11.6^2 = (11.5 + 0.1)^2 = 11.5^2 + 2\times11.5\times0.1 + 0.1^2 = 133.25 + 2.3 + 0.01 = 135.56$

Step4: Approximate to tenths place

Since $133.25 < 134 < 135.56$, and 134 is closer to $11.5^2=133.25$ than to $11.6^2=135.56$.

Answer:

$11.5$