Divide the following expression and simplify completely. Enter your answer as a complex number in standard form. (2+11i)/(3+7i) square

To divide the complex numbers \(\frac{2+11i}{3+7i}\) and simplify completely, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3+7i\) is \(3-7i\).<br /><br />First, let's multiply the numerator and the denominator by the conjugate of the denominator:<br /><br />\[<br />\frac{2+11i}{3+7i} \cdot \frac{3-7i}{3-7i}<br />\]<br /><br />Next, we perform the multiplication in the numerator and the denominator separately:<br /><br />### Numerator:<br />\[<br />(2+11i)(3-7i)<br />\]<br /><br />Using the distributive property (FOIL method):<br />\[<br />= 2 \cdot 3 + 2 \cdot (-7i) + 11i \cdot 3 + 11i \cdot (-7i)<br />\]<br />\[<br />= 6 - 14i + 33i - 77i^2<br />\]<br /><br />Since \(i^2 = -1\):<br />\[<br />-77i^2 = -77(-1) = 77<br />\]<br /><br />So the numerator becomes:<br />\[<br />6 - 14i + 33i + 77 = 83 + 19i<br />\]<br /><br />### Denominator:<br />\[<br />(3+7i)(3-7i)<br />\]<br /><br />Using the difference of squares formula:<br />\[<br />= 3^2 - (7i)^2<br />\]<br />\[<br />= 9 - 49i^2<br />\]<br /><br />Since \(i^2 = -1\):<br />\[<br />-49i^2 = -49(-1) = 49<br />\]<br /><br />So the denominator becomes:<br />\[<br />9 + 49 = 58<br />\]<br /><br />Now we have:<br />\[<br />\frac{83 + 19i}{58}<br />\]<br /><br />We can separate this into real and imaginary parts:<br />\[<br />= \frac{83}{58} + \frac{19i}{58}<br />\]<br /><br />Simplifying each part:<br />\[<br />= \frac{83}{58} + \frac{19}{58}i<br />\]<br /><br />Thus, the simplified form of the given expression is:<br />\[<br />\boxed{\frac{83}{58} + \frac{19}{58}i}<br />\]