To compare \( \sqrt{64} \) and \( \sqrt{59} \), we first calculate \( \sqrt{64} \):
\[ \sqrt{64} = 8 \]
Next, we need to determine \( \sqrt{59} \). We know that \( 59 \) is between \( 49 \) and \( 64 \), which means:
\[ \sqrt{49} < \sqrt{59} < \sqrt{64} \] \[ 7 < \sqrt{59} < 8 \]
This confirms that \( \sqrt{59} \) is less than \( 8 \). We can further refine our approximation by testing squares of numbers between \( 7 \) and \( 8 \).
Let's check \( 7.5 \):
\[ 7.5^2 = 56.25 \]
Now check \( 7.6 \):
\[ 7.6^2 = 57.76 \]
And check \( 7.7 \):
\[ 7.7^2 = 59.29 \]
Since \( 58 < 59 < 59.29 \), we have:
\[ 7.6 < \sqrt{59} < 7.7 \]
This approximation confirms that \( \sqrt{59} \) is greater than \( 7.6 \) but less than \( 7.7 \).
Thus, we can conclude:
\[ \sqrt{64} > \sqrt{59} \]
The inequality statement can be written as:
\[ \sqrt{64} > \sqrt{59} \]
Thus, we fill in the blank to say:
\[ \sqrt{64} > \sqrt{59} \]