Question

Estimate the value √8000

Answer 1

To estimate the value of \( \sqrt{8000} \), we can first simplify \( 8000 \).

We can express \( 8000 \) as:
\[
8000 = 8 \times 1000 = 8 \times 10^3 = (2^3) \times (10^3) = 2^3 \times (10^3) = 2^3 \times (10^{2.5})^2 = 2^3 \times (10^2) \times (10^{0.5})^2
\]

To simplify further:
\[
8000 = 8 \times 1000 = 8 \times (10^3) = 8 \times (1000) = 8 \times (100 \times 10) = 8000 = 8000
\]

Alternatively, we can compute it directly:
\[
\sqrt{8000} = \sqrt{8 \times 1000} = \sqrt{8} \times \sqrt{1000}
\]

Now calculate \( \sqrt{8} \):
\[
\sqrt{8} \approx 2.83
\]

Now calculate \( \sqrt{1000} \):
\[
\sqrt{1000} \approx 31.62
\]

Now multiplying these estimates:
\[
\sqrt{8000} \approx 2.83 \times 31.62 \approx 89.56
\]

Alternatively, we can estimate it more easily:
We know:
\[
\sqrt{8100} = 90 \quad \text{(since } 90^2 = 8100\text{)}
\]
and
\[
\sqrt{6400} = 80 \quad \text{(since } 80^2 = 6400\text{)}
\]

Since \( 8000 \) is between \( 6400 \) and \( 8100 \):
\[
80 < \sqrt{8000} < 90
\]

Estimating a bit closer, we can try \( 89.44^2 = 8000 \):
Continuing this approximation shows it to be closer to 89.44. Thus,
\[
\sqrt{8000} \approx 89.44
\]

So the final estimate is:
\[
\sqrt{8000} \approx 89.44
\]