To solve for the missing value in the given table, let's first examine the nature of the provided numbers. Notice that every number in the table except for the missing value appears to be a perfect square. This suggests that the missing number should also be a perfect square.
Let's write down the perfect squares and their square roots for easier observation:
[tex]\[
\begin{array}{|c|c|c|}
\hline
49 & 64 & 1 \\
\hline
9 & \phantom{?} & 36 \\
\hline
81 & 25 & 16 \\
\hline
\end{array}
\][/tex]
Taking the square roots of these values gives us:
[tex]\[
\begin{array}{|c|c|c|}
\hline
7 & 8 & 1 \\
\hline
3 & ? & 6 \\
\hline
9 & 5 & 4 \\
\hline
\end{array}
\][/tex]
We note that the missing value lies in the middle of the matrix and could correspond to a value that maintains some equilibrium within the pattern of surrounding numbers.
Observing the square roots:
- The left column has values 7, 3, and 9.
- The second column has values 8, ?, and 5.
- The third column has values 1, 6, and 4.
It's not immediately obvious what arithmetic or geometric pattern to follow; therefore, we consider another approach to solve for the unknown √?.
Given the pattern observed:
- Let's take rooted values in a systematic manner. For instance:
- [tex]\( \sqrt{7 \times 8 \times 1 \times 3 \times 6 \times 9 \times 5 \times 4} \)[/tex]
- Doing the multiplication under the square root:
- [tex]\(\sqrt{7 \cdot 8 \cdot 1 \cdot 3 \cdot 6 \cdot 9 \cdot 5 \cdot 4} = \sqrt{362880} \)[/tex]
- Note that since the missing value squared lies in the second row/column intersection:
From the detailed computed estimations provided:
[tex]\[
(\text{Resulting value needs approximation to } ?)
- We derive :
- 2
Thus converting our intact square root
Rough computation indicates that square proximity translates to realistic
Interpretations = number squared,
``Missing Square = 4
Therefore, putting the ultimate value back into table:
The final matrix becomes:
\[
\begin{array}{|c|c|c|}
\hline
49 & 64 & 1 \\
\hline
9 & 4 & 36 \\
\hline
81 & 25 & 16 \\
\hline
\end{array}
\][/tex]
Thus, the missing value is [tex]\( 4 \)[/tex].
