Question: Order the expressions by choosing <, >, or =. (\frac{1}{3})^{-1} \square 3^{-1} 3^{-2} \square…

Order the expressions by choosing <, >, or =. (\frac{1}{3})^{-1} \square 3^{-1} 3^{-2} \square 3^{-1} (\frac{1}{3})^{-2} \square (\frac{1}{3})^{-1}

Solution

To solve the problem, we need to evaluate each expression by applying the exponent rules. Let’s evaluate each pair: 1. Evaluate \(\left(\frac{1}{3}\right)^{-1}\) and \(3^{-1}\): \[ \left(\frac{1}{3}\right)^{-1} = 3 \] \[ 3^{-1} = \frac{1}{3} \] Since \(3 > \frac{1}{3}\), we have: \[ \left(\frac{1}{3}\right)^{-1} > 3^{-1} \] 2. Evaluate \(3^{-2}\) and \(3^{-1}\): \[ 3^{-2} = \frac{1}{9} \] \[ 3^{-1} = \frac{1}{3} \] Since \(\frac{1}{9} < \frac{1}{3}\), we have: \[ 3^{-2} < 3^{-1} \] 3. Evaluate \(\left(\frac{1}{3}\right)^{-2}\) and \(\left(\frac{1}{3}\right)^{-1}\): \[ \left(\frac{1}{3}\right)^{-2} = 9 \] \[ \left(\frac{1}{3}\right)^{-1} = 3 \] Since \(9 > 3\), we have: \[ \left(\frac{1}{3}\right)^{-2} > \left(\frac{1}{3}\right)^{-1} \] Thus, the correct order for each pairing is: 1. \(\left(\frac{1}{3}\right)^{-1} > 3^{-1}\) 2. \(3^{-2} < 3^{-1}\) 3. \(\left(\frac{1}{3}\right)^{-2} > \left(\frac{1}{3}\right)^{-1}\)