Question

Using a number line what is the approximate number value of square root 103 provide your answer and decimal form rounded to the nearest 100 so square root of 103 is approximately what?

Answer 1

To approximate the value of \(\sqrt{103}\) using a number line, we can start by identifying the perfect squares around 103:

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

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

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

Next, we can determine that \(\sqrt{103}\) is closer to \(10\) than to \(11\). To find a more precise approximation, we can try squaring numbers between 10 and 11.

Calculating \(10.1^2\):

\[
10.1^2 = 102.01
\]

Calculating \(10.2^2\):

\[
10.2^2 = 104.04
\]

Since \(102.01 < 103 < 104.04\), we can narrow down \(\sqrt{103}\) to be between \(10.1\) and \(10.2\).

To get even closer, we can calculate \(10.11^2\):

\[
10.11^2 \approx 102.2121
\]

Calculating \(10.12^2\):

\[
10.12^2 \approx 102.4144
\]

Calculating \(10.13^2\):

\[
10.13^2 \approx 102.6169
\]

And \(10.14^2\):

\[
10.14^2 = 102.6196
\]

When you calculate:

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

However, using a calculator, the square root value is approximately:

\[
\sqrt{103} \approx 10.148
\]

Rounding \(10.148\) to the nearest hundredth gives:

\[
\sqrt{103} \approx 10.15
\]

Therefore, rounding to the nearest hundredth, the approximate value of \(\sqrt{103}\) is:

\[
\sqrt{103} \approx 10.15
\]