Sure, let's complete the table and provide a step-by-step explanation for each calculation.
Given the equation: [tex]\( \left( \frac{2}{3} \right)^x \)[/tex]
We need to fill in the values in the table for the specified [tex]\( x \)[/tex] values.
### For [tex]\( x = -1 \)[/tex]:
[tex]\[
\left( \frac{2}{3} \right)^{-1} = \frac{1}{\left( \frac{2}{3} \right)} = \frac{1}{\frac{2}{3}} = \frac{3}{2} = 1.5
\][/tex]
The value when [tex]\( x = -1 \)[/tex] is [tex]\( 1.5 \)[/tex].
### For [tex]\( x = 2 \)[/tex]:
[tex]\[
\left( \frac{2}{3} \right)^{2} = \left( \frac{2}{3} \right) \times \left( \frac{2}{3} \right) = \frac{4}{9} \approx 0.4444444444444444
\][/tex]
The value when [tex]\( x = 2 \)[/tex] is approximately [tex]\( 0.4444444444444444 \)[/tex].
### Complete Table:
Now, we can fill in the values we calculated into the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & \left( \frac{2}{3} \right)^x \\
\hline
-1 & 1.5 \\
\hline
2 & 0.4444444444444444 \\
\hline
\end{array}
\][/tex]
### Final Answer:
The completed table is:
| [tex]\( x \)[/tex] | [tex]\( \left( \frac{2}{3} \right)^x \)[/tex] |
| ------ | ---------- |
| -1 | 1.5 |
| 2 | 0.4444444444444444 |
This completes the step-by-step solution for the given problem.
