Question: Order the expressions by choosing <, >, or =. \[ \begin{array}{ccc} 6^{-2} &…

Order the expressions by choosing <, >, or =.

\[ \begin{array}{ccc} 6^{-2} & \left(\frac{1}{6}\right)^{-1} & \\ \left(\frac{1}{6}\right)^{-1} & \left(\frac{1}{6}\right)^{-2} & \\ \left(\frac{1}{6}\right)^{-2} & 6^{-2} & \end{array} \]

Solution

Find the value of each expression: 1. Evaluate \( \left(6^{-2}\right) \). \[ 6^{-2} = \frac{1}{6^2} = \frac{1}{36} \] 2. Evaluate \( \left(\left(\frac{1}{6}\right)^{-1}\right) \). \[ \left(\frac{1}{6}\right)^{-1} = 6 \] 3. Evaluate \( \left(\left(\frac{1}{6}\right)^{-1}\right) \). \[ \left(\frac{1}{6}\right)^{-1} = 6 \] 4. Evaluate \( \left(\left(\frac{1}{6}\right)^{-2}\right) \). \[ \left(\frac{1}{6}\right)^{-2} = 6^2 = 36 \] 5. Evaluate \( \left(\left(\frac{1}{6}\right)^{-2}\right) \). \[ \left(\frac{1}{6}\right)^{-2} = 36 \] 6. Evaluate \( \left(6^{-2}\right) \). \[ 6^{-2} = \frac{1}{36} \] Compare the values: \[ \begin{cases} \frac{1}{36} < 6 & \\ 6 < 36 & \\ 36 > \frac{1}{36} & \end{cases} \] Order the expressions: \[ \frac{1}{36} < 6 < 36 \]