Question: Ordering numbers with negative exponents Order the expressions by choosing <, >, or =. 7^{-1} <…

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

Solution

Order the expressions by comparing: 1. Evaluate \(7^{-1}\): \[ 7^{-1} = \frac{1}{7} \] 2. Evaluate \(7^{-2}\): \[ 7^{-2} = \frac{1}{7^2} = \frac{1}{49} \] 3. Compare \(7^{-1}\) and \(7^{-2}\): \[ \frac{1}{7} > \frac{1}{49} \] Therefore: \[ 7^{-1} > 7^{-2} \] 4. Evaluate \(\left(\frac{1}{7}\right)^{-1}\): \[ \left(\frac{1}{7}\right)^{-1} = 7 \] 5. Compare \(7^{-1}\) and \(\left(\frac{1}{7}\right)^{-1}\): \[ \frac{1}{7} < 7 \] 6. Evaluate \(7^{-1}\): \[ 7^{-1} = \frac{1}{7} \] Therefore the order is: \[ \begin{cases} (1/7)^{-1} > 7^{-1} > 7^{-2} & \end{cases} \]