Find the exact value of the expression. \[ \sin \left[\sin ^{-1} \frac{3}{5}+\cos ^{-1}\left(-\frac{7}{25}\right)\right] \] \( \boldsymbol{\operatorname { s i n }}\left[\sin ^{-1} \frac{3}{5}+\cos ^{-1}\left(-\frac{7}{25}\right)\right]=\square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

To find the exact value of the expression \[ \sin \left[\sin ^{-1} \frac{3}{5}+\cos ^{-1}\left(-\frac{7}{25}\right)\right], \] we will use the following identities: 1. Let \(\theta = \sin^{-1} \left(\frac{3}{5}\right)\). 2. Therefore, \(\sin(\theta) = \frac{3}{5}\). We can find \(\cos(\theta)\) using the Pythagorean identity: \[ \cos^2(\theta) + \sin^2(\theta) = 1 \implies \cos^2(\theta) + \left(\frac{3}{5}\right)^2 = 1. \] Calculating: \[ \left(\frac{3}{5}\right)^2 = \frac{9}{25} \implies \cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25} \implies \cos(\theta) = \frac{4}{5}. \] Since \(\sin^{-1} \left(\frac{3}{5}\right)\) is in the first quadrant, \(\cos(\theta) = \frac{4}{5}\). Next, let \( \phi = \cos^{-1} \left(-\frac{7}{25}\right) \). Thus, \(\cos(\phi) = -\frac{7}{25}\). To find \(\sin(\phi)\), we apply the Pythagorean identity again: \[ \sin^2(\phi) + \cos^2(\phi) = 1 \implies \sin^2(\phi) + \left(-\frac{7}{25}\right)^2 = 1. \] Calculating: \[ \left(-\frac{7}{25}\right)^2 = \frac{49}{625} \implies \sin^2(\phi) = 1 - \frac{49}{625} = \frac{576}{625}. \] Thus, \[ \sin(\phi) = \sqrt{\frac{576}{625}} = \frac{24}{25}. \] Since \(\cos(\phi)\) is negative, \(\phi\) is in the second quadrant, thus \(\sin(\phi) = \frac{24}{25}\). Now, we can use the angle addition formula for sine: \[ \sin(\theta + \phi) = \sin(\theta)\cos(\phi) + \cos(\theta)\sin(\phi). \] Substituting in the values found: \[ \sin(\theta + \phi) = \left(\frac{3}{5}\right) \left(-\frac{7}{25}\right) + \left(\frac{4}{5}\right) \left(\frac{24}{25}\right). \] Calculating each term: 1. The first term: \[ \frac{3}{5} \cdot -\frac{7}{25} = -\frac{21}{125}. \] 2. The second term: \[ \frac{4}{5} \cdot \frac{24}{25} = \frac{96}{125}. \] Adding these two results together: \[ \sin(\theta + \phi) = -\frac{21}{125} + \frac{96}{125} = \frac{75}{125} = \frac{3}{5}. \] Thus, the value of the expression is: \[ \boxed{\frac{3}{5}}. \]