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} □ 7^{-1} (\frac{1}{7})^{-1} □ (\frac{1}{7})^{-2}

Solution

The task is to order the expressions with negative exponents: 1. \(7^{-1}\) 2. \(7^{-2}\) 3. \(\left(\frac{1}{7}\right)^{-1}\) 4. \(\left(\frac{1}{7}\right)^{-2}\) Let’s simplify each expression: Start with \(7^{-1}\): \[ 7^{-1} = \frac{1}{7} \] Then \(7^{-2}\): \[ 7^{-2} = \frac{1}{7^2} = \frac{1}{49} \] Next, simplify \(\left(\frac{1}{7}\right)^{-1}\): \[ \left(\frac{1}{7}\right)^{-1} = 7 \] Finally, simplify \(\left(\frac{1}{7}\right)^{-2}\): \[ \left(\frac{1}{7}\right)^{-2} = 7^2 = 49 \] Now we order the expressions from least to greatest: \[ 7^{-2} < 7^{-1} < \left(\frac{1}{7}\right)^{-1} < \left(\frac{1}{7}\right)^{-2} \] Thus, the order is: \[ \frac{1}{49} < \frac{1}{7} < 7 < 49 \]