// Numbas version: finer_feedback_settings
{"name": "Utilice las potencias  $i $ para resolver lo siguiente...", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Utilice las potencias  $i $ para resolver lo siguiente...", "tags": [], "metadata": {"description": "<p>Operaciones combinadas con n&uacute;meros complejos.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Utilice las potencias&nbsp; $i $ para resolver y escriba su respuesta en la real-imaginaria ($ a + bi $).</p>", "advice": "<h4>a)</h4>\n<p>Para escribir $z=\\frac{1+i}{1-i}$ en la forma real - imaginaria, multiplicamos $z$ por el complejo conjugado del denominador, obtenemos</p>\n<p>\\[z=\\frac{(1+i)(1+i)}{(1-i)(1+i)}=\\frac{1+2i-1}{1+1}=i.\\]</p>\n<h4>b)</h4>\n<p>La expresi&oacute;n $z=\\left(\\frac{1+i}{1-i}\\right)^\\var{a}$ en forma real - imaginaria,&nbsp; usamos la parte&nbsp;<strong>a)</strong>, de modo que solo necesitamos escribir $ i^\\var{a} $ en forma real-imaginaria.</p>\n<p>El resultado de aumentar $ i $ a un entero arbitrario $n$ se puede determinar usando el hecho de que $i^2=-1$, $i^3=-i$, y $i^4=i$.</p>\n<p>Divida la potenciapor $4$ y calcule el resto (i.e. calcular $n\\mod4$), que puede ser uno de estos valores $0$, $1$, $2$, or $3$.&nbsp;</p>\n<p>Si el resto es $0$, entonces&nbsp;$i^n=1$;</p>\n<p>si el resto es $1$, entonces $i^n=i$;&nbsp;</p>\n<p>si el resto es $2$, entonces $i^n=-1$;&nbsp;</p>\n<p>si el resto es $3$, entonces $i^n=-i$.</p>\n<p>Por lo tanto, $n \\bmod 4 =\\var{moda4}$, so $i^\\var{a}=\\simplify{i^{a}}$.</p>\n<h4>c)</h4>\n<p>Usando el m&eacute;todo de la parte&nbsp;<strong>b)</strong>, $n \\bmod 4=\\var{n} \\bmod 4=\\var{modn4}$, as&iacute; $i^{\\var{n}}=i^{\\var{modn4}}=\\simplify{i^{modn4}}$.</p>", "rulesets": {}, "extensions": [], "variables": {"modn4": {"name": "modn4", 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