Question: Order the expressions by choosing <, >, or =. 7^{-1} □ 7^{-2} (\frac{1}{7})^{-1} □ 7^{-1}…

Order the expressions by choosing <, >, or =. 7^{-1} □ 7^{-2} (\frac{1}{7})^{-1} □ 7^{-1} (\frac{1}{7})^{-1} □ (\frac{1}{7})^{-2}

Solution

To order the expressions, let’s evaluate each one. The expressions are: 1. \( 7^{-1} \) 2. \( 7^{-2} \) 3. \( \left( \frac{1}{7} \right)^{-1} \) 4. \( 7^{-1} \) 5. \( \left( \frac{1}{7} \right)^{-1} \) 6. \( \left( \frac{1}{7} \right)^{-2} \) Evaluate each expression separately: 1. \( 7^{-1} = \frac{1}{7} \) 2. \( 7^{-2} = \frac{1}{7^2} = \frac{1}{49} \) 3. \( \left( \frac{1}{7} \right)^{-1} = 7 \) 4. \( 7^{-1} = \frac{1}{7} \) (Same as expression 1) 5. \( \left( \frac{1}{7} \right)^{-1} = 7 \) (Same as expression 3) 6. \( \left( \frac{1}{7} \right)^{-2} = 7^2 = 49 \) Now, let’s order them from least to greatest: \[ \begin{cases} 7^{-2} = \frac{1}{49} & \\ 7^{-1} = \frac{1}{7} \\ \left( \frac{1}{7} \right)^{-1} = 7 \\ \left( \frac{1}{7} \right)^{-2} = 49 & \end{cases} \] Conclusion: \[ 7^{-2} < 7^{-1} = 7^{-1} < \left( \frac{1}{7} \right)^{-1} = \left( \frac{1}{7} \right)^{-1} < \left( \frac{1}{7} \right)^{-2} \]