{"id":6272,"date":"2020-04-11T18:28:37","date_gmt":"2020-04-11T12:58:37","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=6272"},"modified":"2023-03-09T19:32:01","modified_gmt":"2023-03-09T14:02:01","slug":"surds-in-maths","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/surds-in-maths\/","title":{"rendered":"Surds : Definition, Rules Of Surd,Types"},"content":{"rendered":"\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"560\" height=\"315\" src=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2020\/06\/Surds-in-Maths.png\" alt=\"rules of surds\" class=\"wp-image-6396\" srcset=\"https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2020\/06\/Surds-in-Maths.png 560w, https:\/\/physicscatalyst.com\/article\/wp-content\/uploads\/2020\/06\/Surds-in-Maths-300x169.png 300w\" sizes=\"auto, (max-width: 560px) 100vw, 560px\" \/><\/figure>\n\n\n\n<p>Surds are important topic in Maths. We are going to discuss here Surds Definition,Types of Surds, Rules of Surds, Conjugate surds, Comparison of Surds, rationalizing surds<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">What is Surds in Maths?<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Surds are the nth roots of the positive rational number whose values cannot be determined accurately or basically it can not be expressed as rational number,whole number. So these are irrational numbers.<\/li>\n\n\n\n<li>The nth roots is represented as  $ \\sqrt[n] { } $<\/li>\n\n\n\n<li>$ \\sqrt [n] {}$ is known radical sign. It is also called radic<\/li>\n\n\n\n<li>$\\sqrt [n]{a} $ is also expressed as $a^{1\/n}$<\/li>\n<\/ul>\n\n\n\n<p><strong>Examples<\/strong><\/p>\n\n\n\n<p>$\\sqrt 2$ is a surd<br>$\\sqrt[3] 2$ is a surd<br>$\\sqrt 7$ is a surd<br>$\\sqrt 4$ is not surd as it can expressed as 2<br>$\\sqrt {9\/4} $ is not surd as it can expressed as 3\/2<br>$\\sqrt {\\sqrt 2}$ is also surd as it can be expressed as $\\sqrt[4] {2}$<br>$\\sqrt {2 + \\sqrt 2}$ is not a surd as per the definition but it is not a rational numbers.<\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#f0f5f4\">Get <a href=\"https:\/\/physicscatalyst.com\/article\/surds-questions-with-detailed-solutions\/\">Surds Questions<\/a> with detailed answers and solutions<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Types Of surds<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">(A) Classification based on definition<\/h3>\n\n\n\n<p><strong>Pure Surds<\/strong><br>The surd will only one Rational number is called the Pure Surds<br><em>Example<\/em><br>$\\sqrt 2$ and $\\sqrt 3$ are example of Pure Surds<\/p>\n\n\n\n<p><strong>Mixed Surds<\/strong><br>Surds which are expressed as product of Rational number and Irrational numbers.<br><em>Example<\/em><br>$2 \\sqrt 2$ and $4 \\sqrt 3$ are example of Mixed Surds<\/p>\n\n\n\n<p><strong>How to convert Mixed surds into Pure Surds<\/strong><br>We can move the rational number inside the surd and make it Pure Surds<\/p>\n\n\n\n<p><em>Example<\/em><br>$2 \\sqrt 2$<br>Taking the Rational Number inside the Surd<br>$2 \\sqrt 2= \\sqrt {2^2 \\times 2 } = \\sqrt 8$<\/p>\n\n\n\n<p>So , we need to raise the rational number to the same power as that of root<br><em>More Examples<\/em><br>(a) $2\\sqrt[4] {3} = \\sqrt[4] {2^4 \\times 3 } =\\sqrt[4] { 48}$<br>(b) $5\\sqrt {3} = \\sqrt {5^2 \\times 3 } =\\sqrt { 75}$<br>(c) $3\\sqrt[4] {7} = \\sqrt[4] {3^4 \\times 4 } =\\sqrt[4] { 567}$<br>(d) $2 \\sqrt {5 \\sqrt {2}}= \\sqrt {2^2 \\times 5 \\times \\sqrt {2}} = 2 \\sqrt { \\sqrt {20^2 \\times 2}} = \\sqrt { \\sqrt {800} } = \\sqrt { 800^{1\/2}}= \\sqrt [4] {800}$<\/p>\n\n\n\n<p><strong>How to convert Pure surds into Mixed Surds<\/strong><br>Many times, the rational number inside the surd can be prime factorized and These prime factors can be taken out provided , the exponent of the prime factor is divisible by root power<br><br>Examples<br>(a) $\\sqrt [3] {56} = \\sqrt [3]{ 2^3 \\times 7} = 2 \\sqrt [3] {7}$<br>(b) $\\sqrt [5] {160} = \\sqrt [5] {2^5 \\times 5} = 2 \\sqrt [5] {5}$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">(B) Classification Based on number of terms in it<\/h3>\n\n\n\n<p><strong>(a) Monomial Surd<\/strong><br>A surd having one term only is called Monomial Surd<\/p>\n\n\n\n<p>Example<br>$2 \\sqrt 2$ , $\\sqrt[3] 2$<\/p>\n\n\n\n<p><strong>(b) Binomial Surd<\/strong><br>A surd having two term only is called Binomial Surd<\/p>\n\n\n\n<p>Example<br>$2 \\sqrt 2 + \\sqrt[3] 2$<\/p>\n\n\n\n<p><strong>(c) Trinomial Surd<\/strong><br>A surd having three term only is called Trinomial Surd<\/p>\n\n\n\n<p>$\\sqrt 3 + \\sqrt 2 + \\sqrt 5$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Conjugate Surds<\/h2>\n\n\n\n<p>Two binomial surds diferring only in the sign between sign is called Conjugate Surds<\/p>\n\n\n\n<p>Example<br>$\\sqrt 3 + \\sqrt 2$ and $\\sqrt 3 &#8211; \\sqrt 2$ are Conjugate Surd<br>$\\sqrt 5 + \\sqrt 2$ and $\\sqrt 5 &#8211; \\sqrt 2$ are Conjugate Surd<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Rules of Surds<\/h2>\n\n\n\n<p>1.$(\\sqrt [n] {a})^n = a$<br>Examples<br>$(\\sqrt {2})^2 = 2$<br>$(\\sqrt [3]{3})^3 = 3$<\/p>\n\n\n\n<p>2. $(\\sqrt [n] {a})(\\sqrt [n] {b}) = \\sqrt [n] {ab}$<\/p>\n\n\n\n<p>3. $\\frac {\\sqrt [n] {a}}{ \\sqrt [n] {b}} = \\sqrt [n] {\\frac {a}{b}}$<\/p>\n\n\n\n<p>4. $a \\sqrt c \\pm b \\sqrt c =(a \\pm b) \\sqrt c$<\/p>\n\n\n\n<p>5.  If a and b are both rationals and $\\sqrt {x}$ and $\\sqrt y$ are both surds and<br>$a + \\sqrt x = b + \\sqrt y$ then a = b and x = y<\/p>\n\n\n\n<p>6. $(\\sqrt a + \\sqrt b) (\\sqrt a &#8211; \\sqrt b) = a -b$<\/p>\n\n\n\n<p>7. Rationalizing the Surds. We multiply and divide by the conjugate surds of the denominators<br>$\\frac {1}{ \\sqrt a + \\sqrt b} = \\frac {1}{ \\sqrt a + \\sqrt b} \\times \\frac {\\sqrt a &#8211; \\sqrt b}{ \\sqrt a &#8211; \\sqrt b} = \\frac { \\sqrt a &#8211; \\sqrt b}{ a -b }$<\/p>\n\n\n\n<p>8.  $\\sqrt [n] {a} = a^{1\/n} = a^{m\/mn}= \\sqrt [mn] {a^m}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Comparison Of Surds<\/h2>\n\n\n\n<p>Surds can be compared only when the power of the surds are same.<br>Example<br>$\\sqrt 3 &lt; \\sqrt 5 &lt; \\sqrt 11$<\/p>\n\n\n\n<p>In case the power is different,we need to convert into same power and then perform the comparison<br>Example<br>$\\sqrt [3] 2$ , $\\sqrt [4] {7}$<\/p>\n\n\n\n<p>For converting into same power, we find the LCM of the root power<br>Surd power of $\\sqrt [3] 2$ is 3<br>Surd power of $\\sqrt [4] {7}$ is 4<br>LCM of 3 and 4 is 12<br>Now converting surds into surd power of 12<br>$\\sqrt [3] 2 = \\sqrt [12] { 2^4} = \\sqrt [12] { 16}$<br>$\\sqrt [4] 7 = \\sqrt [12] { 7^3} = \\sqrt [12] { 343}$<br>Now surds can be compared<br>$\\sqrt [12] { 343} &gt; \\sqrt [12] { 16}$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Rationalizing of Surds<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>When we get a rational number by multiplying two surds, then each of the surds is known as rationalizing factors.<\/li>\n\n\n\n<li>So if a surds is given , we can rationalize it by multiplying it another surd. <\/li>\n\n\n\n<li>There can be many surds which can rationalize, we will choose the simplest one always<\/li>\n<\/ul>\n\n\n\n<p>Example<br>$\\sqrt {3}$<br>Now $\\sqrt {3} \\times \\sqrt {3} =3$<br>So $\\sqrt {3}$ is the simplest rationalizing factor<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solved Examples<\/h2>\n\n\n\n<p><strong>1<\/strong>.Rationalize the Denominator<br>$ \\frac {1}{2 &#8211; \\sqrt 3}$<\/p>\n\n\n\n<p><strong>Solution<\/strong><br>$ \\frac {1}{2 &#8211; \\sqrt 3} = \\frac {1}{2 &#8211; \\sqrt 3} \\times \\frac {2 + \\sqrt 3}{2 -\\sqrt 3}= \\frac {2 + \\sqrt 3}{4 -3}= 2 + \\sqrt 3 $<\/p>\n\n\n\n<p><strong>2.<\/strong>  $\\sqrt [3] {32} \\times \\sqrt [3] {16}$<\/p>\n\n\n\n<p><strong>Solution<\/strong><br>$\\sqrt [3] {32} \\times \\sqrt [3] {16} = \\sqrt [3] { 32 \\times 16} = \\sqrt [3] {2^9}= 8$<\/p>\n\n\n\n<p><strong>3.<\/strong> $( \\sqrt 3 + 2 \\sqrt 2) &#8211; ( 3 \\sqrt 2 &#8211; 2 \\sqrt 3)$<\/p>\n\n\n\n<p><strong>Solution<\/strong><br>$( \\sqrt 3 + 2 \\sqrt 2) &#8211; ( 3 \\sqrt 2 &#8211; 2 \\sqrt 3)$<br>$=\\sqrt 3 + 2 \\sqrt 3 + 2 \\sqrt 2 &#8211; 3 \\sqrt 2$<br>$=3 \\sqrt 3 &#8211; \\sqrt 2$<\/p>\n\n\n\n<p>Hope you like the content on Surd Definition, Types of Surds, Rules of Surds, Surds example. Please do provide the feedback<\/p>\n\n\n\n<p><strong>Further References<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Learn about the <a rel=\"noreferrer noopener\" href=\"https:\/\/physicscatalyst.com\/Class9\/numbersystem.php\">Number system for class 9<\/a><\/li>\n\n\n\n<li><a rel=\"noreferrer noopener\" href=\"https:\/\/brilliant.org\/wiki\/surds\/\">https:\/\/brilliant.org\/wiki\/surds\/<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/physicscatalyst.com\/article\/surds-questions-with-detailed-solutions\/\">Surd Questions<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/physicscatalyst.com\/article\/wp-admin\/post.php?post=6314&amp;action=edit\">BODMAS Rule<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/physicscatalyst.com\/article\/wp-admin\/post.php?post=6152&amp;action=edit\">Algebraic Identities<\/a><\/li>\n<\/ul>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Surds are important topic in Maths. We are going to discuss here Surds Definition,Types of Surds, Rules of Surds, Conjugate surds, Comparison of Surds, rationalizing surds What is Surds in Maths? Examples $\\sqrt 2$ is a surd$\\sqrt[3] 2$ is a surd$\\sqrt 7$ is a surd$\\sqrt 4$ is not surd as it can expressed as 2$\\sqrt [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","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":"","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":[498],"tags":[],"class_list":["post-6272","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - 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We are going to discuss here Surds Definition,Types of Surds, Rules of Surds, Conjugate surds, Comparison of Surds, rationalizing surds What is Surds in Maths? Examples $\\sqrt 2$ is a surd$\\sqrt[3] 2$ is a surd$\\sqrt 7$ is a surd$\\sqrt 4$ is not surd as it can expressed as 2$\\sqrt&hellip;","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/6272","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=6272"}],"version-history":[{"count":2,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/6272\/revisions"}],"predecessor-version":[{"id":7711,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/6272\/revisions\/7711"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=6272"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=6272"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=6272"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}