To find the missing number in the table, we'll analyze the pattern present in the [tex]\( y \)[/tex]-values corresponding to each [tex]\( x \)[/tex]-value.
Here is the table we have:
[tex]\[
\begin{tabular}{|l|c|c|c|c|c|}
\hline
$x$ & 1 & 2 & 3 & 4 & 5 \\
\hline
$y$ & 6 & 36 & 216 & ??? & 7,776 \\
\hline
\end{tabular}
\][/tex]
Looking at the [tex]\( y \)[/tex]-values, we notice the following pattern:
1. When [tex]\( x = 1, y = 6 \)[/tex]
2. When [tex]\( x = 2, y = 36 \)[/tex]
3. When [tex]\( x = 3, y = 216 \)[/tex]
4. When [tex]\( x = 5, y = 7776 \)[/tex]
We can see that the [tex]\( y \)[/tex]-values increase rapidly. Given the exponential growth pattern suggested by the values,
allows us to derive a formula for [tex]\( y \)[/tex]:
[tex]\[
y = 6 \cdot 6^{(x - 1)}
\][/tex]
Let's verify this:
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 6 \cdot 6^{(1 - 1)} = 6 \cdot 6^0 = 6 \cdot 1 = 6
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = 6 \cdot 6^{(2 - 1)} = 6 \cdot 6^1 = 6 \cdot 6 = 36
\][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
y = 6 \cdot 6^{(3 - 1)} = 6 \cdot 6^2 = 6 \cdot 36 = 216
\][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[
y = 6 \cdot 6^{(5 - 1)} = 6 \cdot 6^4 = 6 \cdot 1296 = 7776
\][/tex]
Now, to find the value of [tex]\( y \)[/tex] when [tex]\( x = 4 \)[/tex]:
[tex]\[
y = 6 \cdot 6^{(4 - 1)} = 6 \cdot 6^3 = 6 \cdot 216 = 1296
\][/tex]
Thus, the missing number in the table is [tex]\( \boxed{1296} \)[/tex].
