(b) \( \left(\frac{-4}{7}\right)^{-2} \times\left(\frac{-4}{7}\right)^{-3}+\left(\frac{-4}{7}\right)^{1} \)

To solve the expression \( \left(\frac{-4}{7}\right)^{-2} \times\left(\frac{-4}{7}\right)^{-3}+\left(\frac{-4}{7}\right)^{1} \), we first apply the laws of exponents. 1. For \( \left(\frac{-4}{7}\right)^{-2} \times \left(\frac{-4}{7}\right)^{-3} \), we can add the exponents: \[ \left(\frac{-4}{7}\right)^{-2 + (-3)} = \left(\frac{-4}{7}\right)^{-5} \] 2. Now, \( \left(\frac{-4}{7}\right)^{-5} \) can be rewritten as: \[ \frac{1}{\left(\frac{-4}{7}\right)^{5}} = \frac{1}{\frac{(-4)^{5}}{7^{5}}} = \frac{7^{5}}{(-4)^{5}} \] 3. Next, we compute \( \left(\frac{-4}{7}\right)^{1} \): \[ \left(\frac{-4}{7}\right)^{1} = \frac{-4}{7} \] Now we can sum the two results: \[ \frac{7^{5}}{(-4)^{5}} + \frac{-4}{7} \] To compute this, we can calculate \( 7^{5} \) and \( (-4)^{5} \): - \( 7^{5} = 16807 \) - \( (-4)^{5} = -1024 \) Thus we have: \[ \frac{16807}{-1024} + \frac{-4}{7} \] To add these fractions, we need a common denominator. The least common multiple of \(-1024\) and \(7\) is \(-1024 \times 7 = -7168\). Rewriting each fraction: \[ \frac{16807 \cdot 7}{-7168} + \frac{-4 \cdot (-1024)}{-7168} = \frac{117649 - 4096}{-7168} \] Calculating \(117649 - 4096 = 113553\), we have: \[ \frac{113553}{-7168} = -\frac{113553}{7168} \] Therefore, the final result is: \[ -\frac{113553}{7168} \]