Which irrational number lies between the rational numbers 109 and 110? a) 5\u03c0 b) \u221a222 c) 10\u221a3 d) (\u221a2)\u00b3<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n To find an irrational number that lies between the rational numbers 109 and 110, we will evaluate each of the given options: (5\\pi), (\\sqrt{222}), (10\\sqrt{3}), and ((\\sqrt{2})^3).<\/p>\n\n\n\n After evaluating each option, we find that (\\sqrt{222}) is the only number that fits the criteria of being irrational and lying between 109 and 110.<\/p>\n\n\n\n The correct answer is b) (\\sqrt{222})<\/strong>, as it is the only irrational number among the options provided that lies in the specified range. Rational numbers are numbers that can be expressed as a fraction of two integers, while irrational numbers cannot be expressed this way. The approximation and inequalities demonstrate that (\\sqrt{222}) is an irrational number between 109 and 110, fulfilling the problem’s requirements.<\/p>\n","protected":false},"excerpt":{"rendered":" Which irrational number lies between the rational numbers 109 and 110? a) 5\u03c0 b) \u221a222 c) 10\u221a3 d) (\u221a2)\u00b3 The Correct Answer and Explanation is: To find an irrational number that lies between the rational numbers 109 and 110, we will evaluate each of the given options: (5\\pi), (\\sqrt{222}), (10\\sqrt{3}), and ((\\sqrt{2})^3). After evaluating each […]<\/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-157874","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/157874","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=157874"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/157874\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=157874"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=157874"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=157874"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
[
5\\pi \\approx 5 \\times 3.14 = 15.7
]
This value is far greater than 110 and does not lie between 109 and 110.<\/li>\n\n\n\n
To determine if (\\sqrt{222}) is between 109 and 110, we first compute:
[
109^2 = 11881 \\quad \\text{and} \\quad 110^2 = 12100
]
Since (11881 < 222 < 12100), we conclude that:
[
109 < \\sqrt{222} < 110
]
Therefore, (\\sqrt{222}) lies between 109 and 110.<\/li>\n\n\n\n
[
10\\sqrt{3} \\approx 10 \\times 1.732 = 17.32
]
This value is also significantly greater than 110, ruling it out.<\/li>\n\n\n\n
[
(\\sqrt{2})^3 = 2\\sqrt{2} \\approx 2 \\times 1.414 = 2.828
]
This value is also much less than 109, so it does not qualify.<\/li>\n<\/ol>\n\n\n\nConclusion<\/h3>\n\n\n\n