Using rational approximations, what is an inequality statement comparing π and √8? Enter <, >, or =.

Answer 1

To compare \( \pi \) and \( \sqrt{8} \), we can evaluate the numerical values of each.

First, we know that:

\[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \]

We can use the approximate value of \( \sqrt{2} \). A common approximation is:

\[ \sqrt{2} \approx 1.414 \]

Thus, we can calculate \( 2\sqrt{2} \):

\[ \sqrt{8} \approx 2 \times 1.414 = 2.828 \]

Next, we use the known value of \( \pi \):

\[ \pi \approx 3.14159 \]

Now we can compare the two values:

\[ \sqrt{8} \approx 2.828 \] \[ \pi \approx 3.14159 \]

Since \( 2.828 < 3.14159 \), we can conclude that:

\[ \sqrt{8} < \pi \]

Therefore, the inequality statement comparing \( \pi \) and \( \sqrt{8} \) is:

\[ \sqrt{8} < \pi \] or in the required form:

\[ \sqrt{8} < \pi \]