To determine between which two integers the value of [tex]\(\sqrt{89}\)[/tex] lies, we need to consider the properties and behavior of square roots with respect to integers.
1. We start by observing the squares of consecutive integers to establish a range around 89.
- The square of 9 is [tex]\(9^2 = 81\)[/tex].
- The square of 10 is [tex]\(10^2 = 100\)[/tex].
2. Next, we note that 89 falls between these two squares:
[tex]\[
81 < 89 < 100.
\][/tex]
3. Since 81 is the square of 9 and 100 is the square of 10, it means that:
[tex]\[
9 < \sqrt{89} < 10.
\][/tex]
Therefore, [tex]\(\sqrt{89}\)[/tex] lies between the integers 9 and 10.
Hence, the value of [tex]\(\sqrt{89}\)[/tex] is between the two integers:
[tex]\[
\boxed{9 \text{ and } 10}.
\][/tex]
