Which expression is equivalent to i
233
? <\/p>\n\n\n\n
A. 1 <\/p>\n\n\n\n
B. \u20131 <\/p>\n\n\n\n
C. i <\/p>\n\n\n\n
D. \u2013i<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n To find the expression equivalent to ( i^{233} ), we first need to understand the behavior of the powers of ( i ), the imaginary unit defined as ( i = \\sqrt{-1} ). The powers of ( i ) cycle through a specific pattern:<\/p>\n\n\n\n After ( i^4 ), the powers repeat every four steps:<\/p>\n\n\n\n This cyclic pattern shows that the value of ( i^n ) depends on the remainder of ( n ) when divided by 4.<\/p>\n\n\n\n To find ( i^{233} ), we need to determine the remainder of ( 233 \\div 4 ):<\/p>\n\n\n\n This tells us that ( 233 \\equiv 1 \\mod 4 ). According to the established cycle, this means:<\/p>\n\n\n\n [ Thus, the expression ( i^{233} ) simplifies to ( i ).<\/p>\n\n\n\n The correct answer is C. i<\/strong>.<\/p>\n\n\n\n Understanding the powers of ( i ) is essential in complex number operations, particularly in fields like engineering and physics where complex numbers are frequently utilized. By recognizing the repetitive nature of ( i )\u2019s powers, we can simplify expressions involving higher powers without resorting to lengthy calculations. The modulo operation provides a quick method to find the equivalent power. In this case, the calculation reveals that any large exponent can be reduced back to one of the four fundamental values (1, ( i ), -1, or -( i )), thereby streamlining complex number manipulations significantly.<\/p>\n","protected":false},"excerpt":{"rendered":" Which expression is equivalent to i233? A. 1 B. \u20131 C. i D. \u2013i The Correct Answer and Explanation is : To find the expression equivalent to ( i^{233} ), we first need to understand the behavior of the powers of ( i ), the imaginary unit defined as ( i = \\sqrt{-1} ). 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-159231","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/159231","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=159231"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/159231\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=159231"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=159231"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=159231"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
\n
\n
\n
i^{233} = i^{1} = i
]<\/p>\n\n\n\nConclusion<\/h3>\n\n\n\n
Explanation<\/h3>\n\n\n\n