It is given that the mean, median and mode of a data set is $1, 3^x$ and $9^x$ respectively. The possible values of the mode is

Mean, Median, and Mode Relation

Given:

• Mean = 1
• Median = \( 3^x \)
• Mode = \( 9^x \)

Use empirical formula:

\[ \text{Mode} = 3 \cdot \text{Median} - 2 \cdot \text{Mean} \]

\[ 9^x = 3 \cdot 3^x - 2 \Rightarrow (3^x)^2 = 3 \cdot 3^x - 2 \]

Let \( y = 3^x \), then:

\[ y^2 = 3y - 2 \Rightarrow y^2 - 3y + 2 = 0 \Rightarrow (y - 1)(y - 2) = 0 \]

So, \( y = 1 \text{ or } 2 \Rightarrow 9^x = y^2 = 1 \text{ or } 4 \)

✅ Final Answer: \(\boxed{1 \text{ or } 4}\)