30 Calcule: a) (2+i)/(5-3i)(7)/(34)+(11)/(34)i c) (1)/(2+3i)(3)/(13)+(2)/(13)i b) (5+i)/(i),-5i d) (sqrt (3)+i)/(sqrt (3)-i)(1)/(2)+(sqrt (3))/(2)

Para calcular as expressões fornecidas, vamos simplificá-las passo a passo.<br /><br />a) \(\frac{2+i}{5-3i} \cdot \frac{7}{34} + \frac{11}{34}i\)<br /><br />Primeiro, vamos simplificar \(\frac{2+i}{5-3i}\):<br /><br />\[<br />\frac{2+i}{5-3i} \cdot \frac{5+3i}{5+3i} = \frac{(2+i)(5+3i)}{(5-3i)(5+3i)}<br />\]<br /><br />Calculando o numerador:<br /><br />\[<br />(2+i)(5+3i) = 2 \cdot 5 + 2 \cdot 3i + i \cdot 5 + i \cdot 3i = 10 + 6i + 5i + 3i^2 = 10 + 11i + 3(-1) = 10 + 11i - 3 = 7 + 11i<br />\]<br /><br />Calculando o denominador:<br /><br />\[<br />(5-3i)(5+3i) = 5^2 - (3i)^2 = 25 - 9(-1) = 25 + 9 = 34<br />\]<br /><br />Portanto:<br /><br />\[<br />\frac{2+i}{5-3i} = \frac{7 + 11i}{34}<br />\]<br /><br />Agora, multiplicamos por \(\frac{7}{34}\):<br /><br />\[<br />\frac{7 + 11i}{34} \cdot \frac{7}{34} = \frac{49 + 77i}{1156}<br />\]<br /><br />Adicionamos \(\frac{11}{34}i\):<br /><br />\[<br />\frac{49 + 77i}{1156} + \frac{11}{34}i = \frac{49 + 77i}{1156} + \frac{11 \cdot 34}{34 \cdot 34}i = \frac{49 + 77i}{1156} + \frac{374}{1156}i = \frac{49 + 77i + 374i}{1156} = \frac{49 + 451i}{1156}<br />\]<br /><br />Portanto, a resposta é:<br /><br />\[<br />\frac{49 + 451i}{1156}<br />\]<br /><br />b) \(\frac{5+i}{i}, -5i\)<br /><br />Para simplificar \(\frac{5+i}{i}\):<br /><br />\[<br />\frac{5+i}{i} = \frac{5}{i} + \frac{i}{i} = \frac{5}{i} + 1<br />\]<br /><br />Multiplicamos o numerador e o denominador por \(i\):<br /><br />\[<br />\frac{5}{i} = \frac{5 \cdot i}{i \cdot i} = \frac{5i}{-1} = -5i<br />\]<br /><br />Portanto:<br /><br />\[<br />\frac{5+i}{i} = -5i + 1<br />\]<br /><br />A resposta é:<br /><br />\[<br />-5i + 1<br />\]<br /><br />c) \(\frac{1}{2+3i} \cdot \frac{3}{13} + \frac{2}{13}i\)<br /><br />Primeiro, vamos simplificar \(\frac{1}{2+3i}\):<br /><br />\[<br />\frac{1}{2+3i} \cdot \frac{2-3i}{2-3i} = \frac{2-3i}{(2+3i)(2-3i)}<br />\]<br /><br />Calculando o denominador:<br /><br />\[<br />(2+3i)(2-3i) = 2^2 - (3i)^2 = 4 - 9(-1) = 4 + 9 = 13<br />\]<br /><br />Portanto:<br /><br />\[<br />\frac{1}{2+3i} = \frac{2-3i}{13}<br />\]<br /><br />Agora, multiplicamos por \(\frac{3}{13}\):<br /><br />\[<br />\frac{2-3i}{13} \cdot \frac{3}{13} = \frac{6 - 9i}{169}<br />\]<br /><br />Adicionamos \(\frac{2}{13}i\):<br /><br />\[<br />\frac{6 - 9i}{169} + \frac{2}{13}i = \frac{6 - 9i}{169} + \frac{2 \cdot 13}{13 \cdot 13}i = \frac{6 - 9i}{169} + \frac{26}{169}i = \frac{6 - 9i + 26i}{169} = \frac{6 + 17i}{169}<br />\]<br /><br />Portanto, a resposta é:<br /><br />\[<br />\frac{6 + 17