Question

Using a number line, what is
the approximate number value
of radicn number of 103? Provide your
answer in decimal form,
rounded to the nearest
hundredth.

Answer 1

To find the approximate value of the square root of 103, we can start by determining the square roots of nearby perfect squares. The perfect squares around 103 are:

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

Since \(100 < 103 < 121\), we know that:

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

Next, to estimate \(\sqrt{103}\) more closely, we can try squaring some decimal values between 10 and 11:

1. First, let's try \(10.1\):
\[
(10.1)^2 = 102.01
\]
This is less than 103.

2. Next, let's try \(10.2\):
\[
(10.2)^2 = 104.04
\]
This is greater than 103.

Now, we know that:

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

To refine our estimate, we can try a value between 10.1 and 10.2, such as \(10.15\):

\[
(10.15)^2 = 103.0225
\]
This is slightly greater than 103.

Next, try \(10.14\):

\[
(10.14)^2 = 102.8196
\]
This is less than 103.

Now we have:

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

To further narrow it down, we can try \(10.145\):

\[
(10.145)^2 = 103.016025
\]
This is also slightly greater than 103.

Next, try \(10.143\):

\[
(10.143)^2 = 102.887249
\]
This is less than 103.

Now we approximate further and find that:

\[
10.143 < \sqrt{103} < 10.145
\]

For final approximation, we will consider that \(\sqrt{103} \approx 10.14\) rounded to the nearest hundredth.

Thus, the approximate value of \(\sqrt{103}\) in decimal form, rounded to the nearest hundredth, is:

\[
\boxed{10.14}
\]