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

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

Solution

To order the expressions, let’s calculate each one: 1. Calculate \(7^{-1}\) and \(7^{-2}\): \[ 7^{-1} = \frac{1}{7} \] \[ 7^{-2} = \frac{1}{7^2} = \frac{1}{49} \] Since \(\frac{1}{7} > \frac{1}{49}\), we have: \[ 7^{-1} > 7^{-2} \] 2. Calculate \(\left( \frac{1}{7} \right)^{-1}\) and \(\left( \frac{1}{7} \right)^{-2}\): \[ \left( \frac{1}{7} \right)^{-1} = 7 \] \[ \left( \frac{1}{7} \right)^{-2} = 7^2 = 49 \] Since \(7 < 49\), we have: \[ \left( \frac{1}{7} \right)^{-1} < \left( \frac{1}{7} \right)^{-2} \] Therefore, the order of the expressions from left to right are: 1. \(7^{-1} > 7^{-2}\) 2. \(\left( \frac{1}{7} \right)^{-1} < \left( \frac{1}{7} \right)^{-2}\)