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 compare the expressions, we need to simplify each one: 1. Simplify \(7^{-1}\): \[ 7^{-1} = \frac{1}{7} \] 2. Simplify \(7^{-2}\): \[ 7^{-2} = \frac{1}{7^2} = \frac{1}{49} \] 3. Simplify \(\left(\frac{1}{7}\right)^{-1}\): \[ \left(\frac{1}{7}\right)^{-1} = 7 \] 4. Simplify \(\left(\frac{1}{7}\right)^{-2}\): \[ \left(\frac{1}{7}\right)^{-2} = 7^2 = 49 \] 5. Simplify \(7^{-1}\): \[ 7^{-1} = \frac{1}{7} \] Now, we order the expressions: \[ \frac{1}{49} < \frac{1}{7} < \frac{1}{7} < 7 < 49 \] Therefore, the order from smallest to largest is: \[ 7^{-2} < 7^{-1} = \left(\frac{1}{7}\right)^{-1} < \left(\frac{1}{7}\right)^{-2} \] The correct symbols to use are: \[ 7^{-2} < 7^{-1} = \left(\frac{1}{7}\right)^{-1} < \left(\frac{1}{7}\right)^{-2} \]