To compare [tex]\(\sqrt{21}\)[/tex] and [tex]\(\sqrt{17}\)[/tex] using rational approximations, we first need to understand the values of these square roots.
The approximate value of [tex]\(\sqrt{21}\)[/tex] is around 4.5826 and [tex]\(\sqrt{17}\)[/tex] is around 4.1231.
Step-by-step comparison:
1. Determine the value of [tex]\(\sqrt{21}\)[/tex]:
[tex]\[
\sqrt{21} \approx 4.5826
\][/tex]
2. Determine the value of [tex]\(\sqrt{17}\)[/tex]:
[tex]\[
\sqrt{17} \approx 4.1231
\][/tex]
3. Compare the two values:
[tex]\[
4.5826 \quad \text{and} \quad 4.1231
\][/tex]
Since 4.5826 is greater than 4.1231, we can state the inequality as:
[tex]\[
\sqrt{21} > \sqrt{17}
\][/tex]
Thus, the inequality is:
[tex]\[
\sqrt{21} > \sqrt{17}
\][/tex]
