1. We are asked to compare pairs of exponential expressions involving powers of 2 and powers of \( \frac{1}{2} \). The goal is to determine the inequality symbol (\( <, >, = \)) that correctly relates each pair. 2. Recall the important rules: - For any base \( a > 1 \), \( a^m > a^n \) if and only if \( m > n \). - For any base \( 0 < a < 1 \), \( a^m > a^n \) if and only if \( m < n \) because the function is decreasing. 3. Compare \( 2^{13} \) and \( 2^{-13} \): - Since base 2 is greater than 1, the larger exponent corresponds to the larger value. - \( 13 > -13 \), so \( 2^{13} > 2^{-13} \). 4. Compare \( \left(\frac{1}{2}\right)^{13} \) and \( \left(\frac{1}{2}\right)^{-13} \): - Base \( \frac{1}{2} \) is between 0 and 1, so the inequality reverses. - Since \( 13 > -13 \), we have \( \left(\frac{1}{2}\right)^{13} < \left(\frac{1}{2}\right)^{-13} \). 5. Compare \( \left(\frac{1}{2}\right)^{13} \) and \( \left(\frac{1}{2}\right)^{16} \): - Base is between 0 and 1, so larger exponent means smaller value. - Since \( 13 < 16 \), \( \left(\frac{1}{2}\right)^{13} > \left(\frac{1}{2}\right)^{16} \). 6. Compare \( \left(\frac{1}{2}\right)^{-16} \) and \( \left(\frac{1}{2}\right)^{-13} \): - Base is between 0 and 1, so larger exponent means smaller value. - Since \( -16 < -13 \), \( \left(\frac{1}{2}\right)^{-16} > \left(\frac{1}{2}\right)^{-13} \). Final answers: \[ 2^{13} > 2^{-13} \] \[ \left(\frac{1}{2}\right)^{13} < \left(\frac{1}{2}\right)^{-13} \] \[ \left(\frac{1}{2}\right)^{13} > \left(\frac{1}{2}\right)^{16} \] \[ \left(\frac{1}{2}\right)^{-16} > \left(\frac{1}{2}\right)^{-13} \]