Simplify 3-4i\/3+4i + 3+4i\/3-4i<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n To simplify the expression (\\frac{3 – 4i}{3 + 4i} + \\frac{3 + 4i}{3 – 4i}), we will perform the addition of the two complex fractions step by step.<\/p>\n\n\n\n The denominators are (3 + 4i) and (3 – 4i). The common denominator will be the product of these two denominators:<\/p>\n\n\n\n [ Using the difference of squares formula, we can simplify this:<\/p>\n\n\n\n [ Now we rewrite each fraction using the common denominator:<\/p>\n\n\n\n [ Expanding both numerators:<\/p>\n\n\n\n [ [ Now, we add these two results:<\/p>\n\n\n\n [ Putting it all together, we have:<\/p>\n\n\n\n [ Thus, the simplified form of the original expression (\\frac{3 – 4i}{3 + 4i} + \\frac{3 + 4i}{3 – 4i}) is:<\/p>\n\n\n\n [ This result shows that the sum of the two complex fractions has yielded a real number, which illustrates how the imaginary parts canceled out during the addition, resulting in a simpler expression.<\/p>\n","protected":false},"excerpt":{"rendered":" Simplify 3-4i\/3+4i + 3+4i\/3-4i The Correct Answer and Explanation is: To simplify the expression (\\frac{3 – 4i}{3 + 4i} + \\frac{3 + 4i}{3 – 4i}), we will perform the addition of the two complex fractions step by step. Step 1: Find a Common Denominator The denominators are (3 + 4i) and (3 – 4i). The […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-158331","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/158331","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=158331"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/158331\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=158331"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=158331"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=158331"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Find a Common Denominator<\/h3>\n\n\n\n
(3 + 4i)(3 – 4i)
]<\/p>\n\n\n\n
= 3^2 – (4i)^2 = 9 – 16(-1) = 9 + 16 = 25
]<\/p>\n\n\n\nStep 2: Rewrite Each Fraction<\/h3>\n\n\n\n
\\frac{(3 – 4i)(3 – 4i)}{25} + \\frac{(3 + 4i)(3 + 4i)}{25}
]<\/p>\n\n\n\nStep 3: Expand the Numerators<\/h3>\n\n\n\n
\n
= 9 – 12i – 12i + 16i^2 = 9 – 24i – 16 = -7 – 24i
]<\/p>\n\n\n\n\n
= 9 + 12i + 12i + 16i^2 = 9 + 24i – 16 = -7 + 24i
]<\/p>\n\n\n\nStep 4: Combine the Expanded Numerators<\/h3>\n\n\n\n
(-7 – 24i) + (-7 + 24i) = -7 – 7 + (-24i + 24i) = -14 + 0i = -14
]<\/p>\n\n\n\nStep 5: Combine the Whole Expression<\/h3>\n\n\n\n
\\frac{-14}{25}
]<\/p>\n\n\n\nConclusion<\/h3>\n\n\n\n
-\\frac{14}{25}
]<\/p>\n\n\n\n