Question

use the calculator below to approximate \\(\sqrt{32}\\) 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 bounds for $\sqrt{32}$

We know $5^2=25$ and $6^2=36$, so $5 < \sqrt{32} < 6$.

Step2: Test midpoint 5.5

Calculate $5.5^2 = 5.5 \times 5.5 = 30.25$. Since $30.25 < 32$, $\sqrt{32} > 5.5$.

Step3: Test 5.6

Calculate $5.6^2 = 5.6 \times 5.6 = 31.36$. Since $31.36 < 32$, $\sqrt{32} > 5.6$.

Step4: Test 5.7

Calculate $5.7^2 = 5.7 \times 5.7 = 32.49$. Since $32.49 > 32$, $\sqrt{32} < 5.7$. Now we know $5.6 < \sqrt{32} < 5.7$.

Step5: Test midpoint 5.65

Calculate $5.65^2 = 5.65 \times 5.65 = 31.9225$. Since $31.9225 < 32$, $\sqrt{32} > 5.65$.

Step6: Test 5.66

Calculate $5.66^2 = 5.66 \times 5.66 = 32.0356$. Since $32.0356 > 32$, $\sqrt{32} < 5.66$.

Step7: Compare differences

Difference between 32 and $5.65^2$: $32 - 31.9225 = 0.0775$
Difference between $5.66^2$ and 32: $32.0356 - 32 = 0.0356$
The value 5.66 is closer to $\sqrt{32}$.

Answer:

$\sqrt{32} \approx 5.66$