Question: Order the expressions by choosing <, >, or =. 5^{-1} □ 5^{-2} 5^{-1} □ (\frac{1}{5})^{-1} 5^{-2} □…

Order the expressions by choosing <, >, or =. 5^{-1} □ 5^{-2} 5^{-1} □ (\frac{1}{5})^{-1} 5^{-2} □ (\frac{1}{5})^{-2}

Solution

To solve the problem, let’s evaluate each expression with negative exponents and compare them. 1. Evaluate \(5^{-1}\) and \(5^{-2}\): \[ 5^{-1} = \frac{1}{5} \] \[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \] Since \(\frac{1}{5} > \frac{1}{25}\), \[ 5^{-1} > 5^{-2} \] 2. Evaluate \(5^{-1}\) and \(\left(\frac{1}{5}\right)^{-1}\): We already know: \[ 5^{-1} = \frac{1}{5} \] Now, evaluate \(\left(\frac{1}{5}\right)^{-1}\): \[ \left(\frac{1}{5}\right)^{-1} = 5 \] Since \(\frac{1}{5} < 5\), \[ 5^{-1} < \left(\frac{1}{5}\right)^{-1} \] 3. Evaluate \(5^{-2}\) and \(\left(\frac{1}{5}\right)^{-2}\): We already know: \[ 5^{-2} = \frac{1}{25} \] Now, evaluate \(\left(\frac{1}{5}\right)^{-2}\): \[ \left(\frac{1}{5}\right)^{-2} = 5^2 = 25 \] Since \(\frac{1}{25} < 25\), \[ 5^{-2} < \left(\frac{1}{5}\right)^{-2} \] The correct order is: \(5^{-1} > 5^{-2}\), \(5^{-1} < \left(\frac{1}{5}\right)^{-1}\), and \(5^{-2} < \left(\frac{1}{5}\right)^{-2}\).