Question: Order the expressions by choosing <, >, or =. 9^{-2} [ ] (\frac{1}{9})^{-2} 9^{-1} [ ]…

Order the expressions by choosing <, >, or =. 9^{-2} [ ] (\frac{1}{9})^{-2} 9^{-1} [ ] (\frac{1}{9})^{-1} 9^{-2} [ ] (\frac{1}{9})^{-1}

Solution

To solve the problem, we need to compare the expressions: 1. \( 9^{-2} \) 2. \( \left( \frac{1}{9} \right)^{-2} \) 3. \( 9^{-1} \) 4. \( \left( \frac{1}{9} \right)^{-1} \) 5. \( 9^{-2} \) 6. \( \left( \frac{1}{9} \right)^{-1} \) First, simplify each expression: For \( 9^{-2} \): \[ 9^{-2} = \frac{1}{9^2} = \frac{1}{81} \] For \( \left( \frac{1}{9} \right)^{-2} \): \[ \left( \frac{1}{9} \right)^{-2} = 9^2 = 81 \] For \( 9^{-1} \): \[ 9^{-1} = \frac{1}{9} \] For \( \left( \frac{1}{9} \right)^{-1} \): \[ \left( \frac{1}{9} \right)^{-1} = 9 \] Now, compare the simplified values: - \( \frac{1}{81} \) - \( 81 \) - \( \frac{1}{9} \) - \( 9 \) Ordering from smallest to largest: 1. \( \frac{1}{81} \) 2. \( \frac{1}{9} \) 3. \( 9 \) 4. \( 81 \) Since expressions 1, 5 are \( \frac{1}{81} \), expressions 2, 6 are \( 9 \), expression 3 is \( \frac{1}{9} \), and expression 4 is \( 81 \), we have: \[ 9^{-2} < 9^{-1} < \left( \frac{1}{9} \right)^{-1} < \left( \frac{1}{9} \right)^{-2} \]