{"id":263359,"date":"2025-07-21T06:01:02","date_gmt":"2025-07-21T06:01:02","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=263359"},"modified":"2025-07-21T06:01:05","modified_gmt":"2025-07-21T06:01:05","slug":"write-the-decimal-expansion-of-root-2-root-3-root-5","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/21\/write-the-decimal-expansion-of-root-2-root-3-root-5\/","title":{"rendered":"write the decimal expansion of root 2 , root 3 , root 5"},"content":{"rendered":"\n<p>write the decimal expansion of root 2 , root 3 , root 5<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The decimal expansions of the square roots of 2, 3, and 5 are as follows:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u221a2 \u2248 1.414213562<\/strong><\/li>\n\n\n\n<li><strong>\u221a3 \u2248 1.732050808<\/strong><\/li>\n\n\n\n<li><strong>\u221a5 \u2248 2.236067977<\/strong><\/li>\n<\/ol>\n\n\n\n<p>These values are <strong>irrational numbers<\/strong>, meaning they cannot be expressed as a finite decimal or a simple fraction. Their decimal expansions go on indefinitely without repeating.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The square roots of numbers like 2, 3, and 5 are <strong>irrational<\/strong> because there are no exact fractions that represent their values. When attempting to calculate their square roots, the result is a non-repeating and non-terminating decimal. For example:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u221a2<\/strong>: The square root of 2 is approximately <strong>1.414213562<\/strong>. This value was famously proven to be irrational by the ancient Greeks using a method of contradiction. The decimal never ends or repeats.<\/li>\n\n\n\n<li><strong>\u221a3<\/strong>: Similarly, the square root of 3 is approximately <strong>1.732050808<\/strong>, and like \u221a2, it also has a non-repeating, non-terminating decimal. The irrationality of \u221a3 was established in much the same way.<\/li>\n\n\n\n<li><strong>\u221a5<\/strong>: The square root of 5 is approximately <strong>2.236067977<\/strong>. It also cannot be expressed as a fraction of two integers and has an infinite decimal expansion.<\/li>\n<\/ul>\n\n\n\n<p>Mathematically, an irrational number cannot be accurately written down in decimal form beyond a certain number of digits. However, these approximations are commonly used when precise calculations are not required. To get more digits for more accuracy, one would need to use a calculator or a numerical method to compute the square roots to the desired precision.<\/p>\n\n\n\n<p>These irrational square roots are essential in many areas of mathematics and physics, and they often appear in geometric problems, such as those involving the Pythagorean theorem.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>write the decimal expansion of root 2 , root 3 , root 5 The correct answer and explanation is: The decimal expansions of the square roots of 2, 3, and 5 are as follows: These values are irrational numbers, meaning they cannot be expressed as a finite decimal or a simple fraction. Their decimal expansions [&hellip;]<\/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-263359","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/263359","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=263359"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/263359\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=263359"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=263359"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=263359"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}