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Let's analyze the given data table to determine if it shows exponential growth, exponential decay, or neither.

The table provided is:

\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 2 & 3 & 4 & 5 & 6 \\
\hline
y & 7,776 & 1,296 & 216 & 36 & 6 \\
\hline
\end{array}
\]

### To check for exponential behavior, let's look for a constant ratio between consecutive \( y \)-values.

- From \( y_2 = 7,776 \) to \( y_3 = 1,296 \):
  \[
  \frac{y_3}{y_2} = \frac{1,296}{7,776} = \frac{1}{6}
  \]

- From \( y_3 = 1,296 \) to \( y_4 = 216 \):
  \[
  \frac{y_4}{y_3} = \frac{216}{1,296} = \frac{1}{6}
  \]

- From \( y_4 = 216 \) to \( y_5 = 36 \):
  \[
  \frac{y_5}{y_4} = \frac{36}{216} = \frac{1}{6}
  \]

- From \( y_5 = 36 \) to \( y_6 = 6 \):
  \[
  \frac{y_6}{y_5} = \frac{6}{36} = \frac{1}{6}
  \]

### Conclusion:
Since the ratio between consecutive \( y \)-values is consistently \( \frac{1}{6} \), the data exhibits **exponential decay**.

Would you like more details on how exponential decay is determined, or do you have any other questions?

### Related Questions:
1. How do you differentiate between exponential growth and decay?
2. What is the general form of an exponential decay function?
3. Can exponential decay apply to real-world phenomena like radioactive decay?
4. How does exponential decay compare with linear decay?
5. What happens to \( y \) values as \( x \) increases in an exponential decay function?

### Tip:
In exponential decay, the common ratio between consecutive values is always less than 1, which causes the values to decrease rapidly over time.

Determine if the data in the table of values shows exponential growth, exponential decay, or neither.

Learn how to determine whether a given data table shows exponential growth, exponential decay, or neither. Using a step-by-step process, the problem involves checking for a constant ratio between consecutive y-values, with an example table showing data where y-values decrease by a factor of 1/6.

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