Question

Approximate square root of 103, put in decimal form rounded to the nearest hundredth

Answer 1

To approximate the square root of 103, we can start by estimating between the squares of two whole numbers.

We know that:

- \(10^2 = 100\)
- \(11^2 = 121\)

Since \(103\) is between \(100\) and \(121\), we know that:

\[
10 < \sqrt{103} < 11
\]

Next, we can get closer by testing values between 10 and 11.

Calculating a few decimal values, we find:

- \(10.1^2 = 102.01\)
- \(10.2^2 = 104.04\)

Since \(102.01 < 103 < 104.04\), we know:

\[
10.1 < \sqrt{103} < 10.2
\]

Let's narrow it down further:

- \(10.15^2 = 103.0225\)

Since \(103.0225 > 103\), we also know:

\[
10.1 < \sqrt{103} < 10.15
\]

Next, let's test \(10.14\):

- \(10.14^2 = 102.8196\)

Since \(102.8196 < 103\), we have:

\[
10.14 < \sqrt{103} < 10.15
\]

Thus, we can approximate \(\sqrt{103} \approx 10.14\).

Now, rounding to the nearest hundredth gives us:

\[
\sqrt{103} \approx 10.14
\]