To compare \(\sqrt{64}\) and \(\sqrt{59}\) using rational approximations, we can start by calculating the values of these square roots.
-
We know that: \[ \sqrt{64} = 8 \]
-
For \(\sqrt{59}\), we can use a rational approximation:
- We know \(\sqrt{49} = 7\) and \(\sqrt{64} = 8\), so \(\sqrt{59}\) is between 7 and 8.
- To get a more accurate approximation, we can estimate by checking perfect squares around 59:
- \( 7.5^2 = 56.25 \)
- \( 7.6^2 = 57.76 \)
- \( 7.7^2 = 59.29 \)
Since \( 7.6^2 < 59 < 7.7^2 \), we can conclude that: \[ 7.6 < \sqrt{59} < 7.7 \]
-
Hence, we can place \(\sqrt{59}\) approximately between 7.6 and 7.7.
Now we can compare: \[ \sqrt{64} = 8 \quad \text{and} \quad 7.6 < \sqrt{59} < 7.7 \]
Thus, the inequality statement comparing \(\sqrt{64}\) and \(\sqrt{59}\) is: \[ \sqrt{64} > \sqrt{59} \]
The final inequality statement is: \[ 8 > \sqrt{59} \]