{"id":13272,"date":"2020-12-10T09:20:31","date_gmt":"2020-12-10T03:50:31","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=13272"},"modified":"2020-12-10T10:56:27","modified_gmt":"2020-12-10T05:26:27","slug":"surds","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/surds\/","title":{"rendered":"Surds"},"content":{"rendered":"<h2>Surds<\/h2>\n<p><iframe loading=\"lazy\" title=\"What are Surds? | Don&#039;t Memorise\" width=\"1200\" height=\"675\" src=\"https:\/\/www.youtube.com\/embed\/568dGLFTom8?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/p>\n<h3><strong>Definition of surd<\/strong><\/h3>\n<p>Any root of a number which can not be exactly found is called a <strong>surd<\/strong>.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-128752\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Surds-1.jpg\" alt=\"Surds 1\" width=\"700\" height=\"425\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Surds-1.jpg 700w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Surds-1-300x182.jpg 300w\" sizes=\"(max-width: 700px) 100vw, 700px\" \/><\/p>\n<p>Let a be a rational number and n is a positive integer. If the n<sup>th\u00a0<\/sup>root of x i.e., x<sup>1\/n<\/sup>\u00a0is irrational, then it is called surd of order n.<\/p>\n<p>Order of a surd is indicated by the number denoting the root.<\/p>\n<p>For example, \\(\\sqrt { 7 } ,\\sqrt [ 3 ]{ 9 } ,{ 11 }^{ 3\/5 },\\sqrt [ n ]{ 3 }\u00a0\\) are surds of second, third, fifth and n<sup>th<\/sup> order respectively.<\/p>\n<p>A second order surd is often called a <strong>quadratic surd<\/strong>, a surd of third order is called a <strong>cubic surd<\/strong>.<\/p>\n<p><strong>Read More:<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/cbselibrary.com\/exponent\/\">What is an Exponent?<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/laws-exponents\/\">What are Laws of Exponents?<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/review-of-exponents\/\">Review of Exponents<\/a><\/li>\n<\/ul>\n<p>https:\/\/www.youtube.com\/watch?v=jyN9U81Z88A<\/p>\n<h3>Types of surds<\/h3>\n<ol>\n<li><strong>Simple surds:<\/strong> A surd consisting of a single term. For example 2\u221a3, 6\u221a5, \u221a5 etc.<\/li>\n<li><strong>Pure and mixed surds:<\/strong> A surd consisting of wholly of an irrational number is called pure surd.<br \/>\nA surd consisting of the product of a rational number and an irrational number is called a mixed surd.<\/li>\n<li><strong>Compound surds:<\/strong> An expression consisting of the sum or difference of two or more surds.<\/li>\n<li><strong>Similar surds:<\/strong> If the surds are different multiples of the same surd, they are called similar surds.<\/li>\n<li><strong>Binomial surds:<\/strong> A compound surd consisting of two surds is called a binomial surd.<\/li>\n<li><strong>Binomial quadratic surds:<\/strong> Binomial surds consisting of pure (or simple) surds of order two i.e., the surds of the form a\u221ab \u00b1 c\u221ad or a \u00b1 b\u221ac are called binomial quadratic surds.<br \/>\nTwo binomial quadratic surds which differ only in the sign which connects their terms are said to be conjugate or complementary to each other. The product of a binomial quadratic surd and its conjugate is always rational.<br \/>\nFor example: The conjugate of the surd 2\u221a7 + 5\u221a3 is the surd 2\u221a7 &#8211;\u00a05\u221a3.<\/li>\n<\/ol>\n<h3>Properties of quadratic surds<\/h3>\n<ol>\n<li>The square root of a rational number cannot be expressed as the sum or difference of a rational number and a quadratic surd.<\/li>\n<li>If two quadratic surds cannot be reduced to others, which have not the same irrational part, their product is irrational.<\/li>\n<li>One quadratic surd cannot be equal to the sum or difference of two others, not having the same irrational part.<\/li>\n<li>If a + \u221ab = c + \u221ad where a and c are rational, and \u221ab, \u221ad \u00a0are irrational, then a = c and b = d.<\/li>\n<\/ol>\n<h3>Rationalisation factors<\/h3>\n<p>If two surds be such that their product is rational, then each one of them is called rationalising factor of the other.<br \/>\nThus each of 2\u221a3 and \u221a3\u00a0is a rationalising factor of each other. Similarly \u221a3 + \u221a2 and \u221a3 &#8211; \u221a2 are rationalising factors of each other, as (\u221a3 + \u221a2)(\u221a3 &#8211; \u221a2) = 1, which is rational.<br \/>\nTo find the factor which will rationalize any given binomial surd:<br \/>\n<strong>Case I:<\/strong> Suppose the given surd is \\(\\sqrt [ p ]{ a } -\\sqrt [ q ]{ b } \\)<br \/>\nLet a<sup>1\/p <\/sup>= x, b<sup>1\/q <\/sup>= y \u00a0and let n be the L.C.M. of p and q. Then x<sup>n<\/sup> and y<sup>n<\/sup> are both rational.<br \/>\nNow x<sup>n<\/sup> &#8211; y<sup>n<\/sup> is divisible by\u00a0x &#8211; y for all values of n, and x<sup>n<\/sup> &#8211; y<sup>n<\/sup> = (x \u2013 y)(x<sup>n-1<\/sup> + x<sup>n-2<\/sup>y + x<sup>n-3<\/sup>y<sup>2<\/sup> + \u2026.. + y<sup>n-1<\/sup>).<br \/>\nThus the rationalizing factor is x<sup>n-1<\/sup> + x<sup>n-2<\/sup>y + x<sup>n-3<\/sup>y<sup>2<\/sup> + \u2026.. + y<sup>n-1<\/sup>\u00a0and the rational product is x<sup>n<\/sup> &#8211; y<sup>n<\/sup>.<br \/>\n<strong>Case II:<\/strong> Let the given surd be \\(\\sqrt [ p ]{ a } +\\sqrt [ q ]{ b } \\)<br \/>\nLet have the same meaning as in Case I.<br \/>\n(1) If n is even, then x<sup>n<\/sup> &#8211; y<sup>n<\/sup>\u00a0is divisible by x + y and\u00a0x<sup>n<\/sup> &#8211; y<sup>n<\/sup> = (x +\u00a0y)(x<sup>n-1<\/sup>\u00a0&#8211; x<sup>n-2<\/sup>y +\u00a0x<sup>n-3<\/sup>y<sup>2<\/sup>\u00a0&#8211; \u2026.. &#8211;\u00a0y<sup>n-1<\/sup>).<br \/>\nThus the rationalizing factor is x<sup>n-1<\/sup>\u00a0&#8211; x<sup>n-2<\/sup>y +\u00a0x<sup>n-3<\/sup>y<sup>2<\/sup>\u00a0&#8211; \u2026.. &#8211;\u00a0y<sup>n-1<\/sup>\u00a0and the rational product is x<sup>n<\/sup> &#8211; y<sup>n<\/sup>.<br \/>\n(2) If n is odd, x<sup>n<\/sup>\u00a0+ y<sup>n<\/sup>\u00a0is divisible by x + y and\u00a0x<sup>n<\/sup>\u00a0+ y<sup>n<\/sup> = (x +\u00a0y)(x<sup>n-1<\/sup>\u00a0&#8211; x<sup>n-2<\/sup>y +\u00a0x<sup>n-3<\/sup>y<sup>2<\/sup>\u00a0&#8211; \u2026.. + y<sup>n-1<\/sup>)<br \/>\nThus the rationalizing factor is and the rational product is x<sup>n<\/sup>\u00a0+ y<sup>n<\/sup>.<\/p>\n<h3>Square roots of a +\u221ab and a + \u221ab + \u221ac + \u221ad where \u221ab , \u221ac , \u221ad are Surds<\/h3>\n<p>Let \u00a0\u221a(\u221aa + \u221ab) = \u221ax + \u221ay where x, y &gt; 0 are rational numbers.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-128753\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Surds-2.jpg\" alt=\"Surds 2\" width=\"723\" height=\"331\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Surds-2.jpg 723w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Surds-2-300x137.jpg 300w\" sizes=\"(max-width: 723px) 100vw, 723px\" \/><br \/>\nThen by squaring and equating, we get equations in x, y, z. On solving these equations, we can find the required square roots.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-128755\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Surds-3.jpg\" alt=\"Surds 3\" width=\"600\" height=\"622\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Surds-3.jpg 600w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Surds-3-289x300.jpg 289w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<h3>Cube root of a binomial quadratic surd<\/h3>\n<h3><img loading=\"lazy\" class=\"alignnone size-full wp-image-128756\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Surds-4.jpg\" alt=\"Surds 4\" width=\"599\" height=\"555\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Surds-4.jpg 599w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/05\/Surds-4-300x278.jpg 300w\" sizes=\"(max-width: 599px) 100vw, 599px\" \/><br \/>\nEquations involving surds<\/h3>\n<p>While solving equations involving surds, usually we have to square, on squaring the domain of the equation extends and we may get some extraneous solutions, and so we must verify the solutions and neglect those which do not satisfy the equation.<br \/>\nNote that from ax = bx, to conclude a = b is not correct. The correct procedure is x(a &#8211; b)=0 i.e. x = 0 or a = b. Here, necessity of verification is required.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Surds Definition of surd Any root of a number which can not be exactly found is called a surd. Let a be a rational number and n is a positive integer. If the nth\u00a0root of x i.e., x1\/n\u00a0is irrational, then it is called surd of order n. 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