{"id":1933,"date":"2023-10-02T07:27:02","date_gmt":"2023-10-02T07:27:02","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=1933"},"modified":"2023-10-02T07:28:26","modified_gmt":"2023-10-02T07:28:26","slug":"week-9-ss1-first-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-9-ss1-first-term-further-mathematics-notes\/","title":{"rendered":"Week 9 &#8211; SS1 First Term Further Mathematics Notes"},"content":{"rendered":"<p>\u00a0<br \/>\n\u00a0<strong>WEEK NINE<br \/>\n<\/strong><strong>TOPIC:  SURDS<br \/>\n<\/strong><strong>CONTENT<br \/>\n<\/strong><\/p>\n<ul>\n<li>\n<div>Rules of surds\n<\/div>\n<\/li>\n<li>\n<div>Basic Form of Surds\n<\/div>\n<\/li>\n<li>\n<div>Similar Surds\n<\/div>\n<\/li>\n<li>\n<div>Conjugate Surds\n<\/div>\n<\/li>\n<li>\n<div>Simplification of Surds\n<\/div>\n<\/li>\n<li>\n<div>Additional &amp; Subtraction of Surds\n<\/div>\n<\/li>\n<li>\n<div>Multiplication and Division of Surds\n<\/div>\n<\/li>\n<li>\n<div>Rationalization of Surds\n<\/div>\n<\/li>\n<li>\n<div>Equality of Surds\n<\/div>\n<\/li>\n<\/ul>\n<p><strong>Rules of Surds<br \/>\n<\/strong> Surds are irrational numbers. They are the root of rational numbers whose value cannot be expressed as exact fractions. Examples of surds are: \u221a2, \u221a7, \u221a12, \u221a18, etc.<\/p>\n<ol>\n<li>\n<div>\u221a(a X b ) = \u221aa X \u221a b\n<\/div>\n<\/li>\n<li>\n<div>\u221a(a \/ b )  = \u221aa  \/ \u221ab\n<\/div>\n<\/li>\n<li>\n<div>\u221a(a + b ) \u2260 \u221aa +  \u221ab\n<\/div>\n<\/li>\n<li>\n<div>\u221a(a \u2013 b ) \u2260 \u221aa &#8211;  \u221ab\n<\/div>\n<\/li>\n<\/ol>\n<p><strong>Basic Forms of Surds<br \/>\n<\/strong>      \u221aa is said to be in its basic form if A does not have a factor that is a perfect square. E.g.  \u221a6, \u221a5, \u221a3, \u221a2 etc.  \u221a18 is not in its basic form because it can be broken into \u221a (9&#215;2) = 3\u221a2. Hence 3\u221a2 is now in its basic form. <\/p>\n<p>\u00a0<strong>Similar Surds<br \/>\n<\/strong>     Surds are similar if their irrational part contains the same numerals e.g. <\/p>\n<ol>\n<li>\n<div>3\u221an and 5\u221an\n<\/div>\n<\/li>\n<li>\n<div>6\u221a2 and 7\u221a2\n<\/div>\n<\/li>\n<\/ol>\n<p>\u00a0<strong>Conjugate Surds<br \/>\n<\/strong>Conjugate surds are two surds whose product result is a rational number.<br \/>\n (i)The conjugate of \u221a3 &#8211; \u221a5 is \u221a3 + \u221a5<br \/>\n     The conjugate of -2\u221a7 + \u221a3 is 2\u221a7 &#8211; \u221a3<br \/>\n     In general, the conjugate of \u221ax + \u221ay is \u221ax &#8211; \u221ay<br \/>\n     The conjugate of \u221ax &#8211; \u221ay = \u221ax + \u221ay <\/p>\n<p>\u00a0<strong>Simplification of Surds<br \/>\n<\/strong>   Surds can be simplified either in the basic form or as a single surd. <\/p>\n<p>\u00a0<strong>Examples<br \/>\n<\/strong> Simplify the following in its basic form (a) \u221a45 (b) \u221a98 <\/p>\n<p>\u00a0<strong>Solution<br \/>\n<\/strong>(a) \u221a45 = \u221a (9 x 5) = \u221a9 x \u221a5 = 3\u221a5<br \/>\n(b) \u221a98 = \u221a (49 x 2) = \u221a49 x \u221a2 = 7\u221a2 <\/p>\n<p>\u00a0<strong>Examples<br \/>\n<\/strong>Simplify the following as a single surd (a) 2\u221a5 (b) 17\u221a2<\/p>\n<p>\u00a0<strong>Solution<br \/>\n<\/strong>(a) 2\u221a5 = \u221a4 x \u221a5 = \u221a (4 x 5) = \u221a20<br \/>\n(b) 17\u221a2 = \u221a289 x \u221a2 = \u221a (289 x 2) = \u221a578<br \/>\n<strong>Addition and Subtraction of Surds<br \/>\n<\/strong> Surds in their basic forms which are similar can be added or subtracted. <\/p>\n<p>\u00a0<strong>Examples<\/strong><br \/>\n\t\tEvaluate the following<br \/>\n(a)\u221a32 + 3\u221a8       (b) 7\u221a3 &#8211; \u221a75      (c) 3\u221a48 &#8211; \u221a75 + 2\u221a12<br \/>\nSolution<\/p>\n<ol>\n<li>\n<div> (\u221a32  + 3\u221a8\n<\/div>\n<\/li>\n<\/ol>\n<p>          = \u221a (16 x 2) + 3\u221a (4 x 2)<br \/>\n          =4\u221a2 + 6\u221a2<br \/>\n          = 10\u221a2 <\/p>\n<p>\u00a0     (b) 7\u221a3 &#8211; \u221a75<br \/>\n= 7\u221a3 &#8211; \u221a (25 x 3)<br \/>\n=7\u221a3 \u2013 5\u221a3     =2\u221a2<br \/>\n     (c) 3\u221a48 &#8211; \u221a75 + 2\u221a12<br \/>\n          = 3\u221a (16 x 3) &#8211; \u221a (25 x 3) + 2\u221a (4 x 3)<br \/>\n          = 12\u221a3 &#8211; 5\u221a3 + 4\u221a3<br \/>\n          = 11\u221a3 <\/p>\n<p>\u00a0<strong>Evaluation<br \/>\n<\/strong>1. Simplify the following <strong>(a) <\/strong>5\u221a 12 &#8211; 3\u221a 18 + 4\u221a72 + 2\u221a75     (b) 3\u221a2 &#8211; \u221a32 + \u221a50 + \u221a98<br \/>\n2. Simplify the following as a single surd (i) 8\u221a3      (ii) 13\u221a2<\/p>\n<p>\u00a0<strong>Multiplication and Division of Surds<br \/>\n<\/strong>Example: Evaluate the following (a) \u221a45 x \u221a28    (b) \u221a24 \/\u221a50 <\/p>\n<p>\u00a0<strong>Solution<br \/>\n<\/strong>(a) \u221a45 x \u221a28<br \/>\n= \u221a (9 x 5) x \u221a (4 x 7)<br \/>\n= 3\u221a5 x 2\u221a7<br \/>\n= 3 x 2 x \u221a (5 x 7)<br \/>\n= 6\u221a35<br \/>\n(b)\u221a24 \/ \u221a50<br \/>\n          = \u221a (24 \/ 50)<br \/>\n          = \u221a (12 \/ 25)<br \/>\n          = \u221a12 \/ \u221a25<br \/>\n          = \u221a (4 x 3) \/ 5<br \/>\n          = 2\u221a3 \/ 5 <\/p>\n<p>\u00a0<strong>Evaluation<\/strong>:<br \/>\n Simplify 1. \u221a6 x (3 &#8211; \u221a5)      2. (2\u221a3 &#8211; \u221a7)(2\u221a3 + \u221a7)<br \/>\n                2. Multiply the following by their conjugate (a) \u221a3 &#8211; 2\u221a5 (b) 3\u221a2 + 2\u221a3<\/p>\n<p>\u00a0<strong>Surds Rationalisation<br \/>\n<\/strong>Rationalisation of surds means multiplying the numerator and denominator by the denominator or by the conjugate of the denominator. <\/p>\n<ol>\n<li>\n<div>Example: Evaluate the following (a) 6\/\u221a3   (b)     \u00a0\u00a0\u00a0\u00a0   3\n<\/div>\n<\/li>\n<\/ol>\n<p><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week9SS1Fi1.png\" alt=\"\"\/>                                                                                             \u221a3 + \u221a2 <\/p>\n<p>\u00a0<br \/>\n\u00a0<strong>Solution<br \/>\n<\/strong><\/p>\n<ol>\n<li>\n<div>6\/\u221a3<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week9SS1Fi2.png\" alt=\"\"\/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0                (b)            3\n<\/div>\n<\/li>\n<\/ol>\n<p><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week9SS1Fi3.png\" alt=\"\"\/>=   6 x \u221a3                                                                                     \u221a3 + \u221a2<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week9SS1Fi4.png\" alt=\"\"\/>  \u221a3 x \u221a3                                                                           =       3 (\u221a3 &#8211; \u221a2)<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week9SS1Fi5.png\" alt=\"\"\/>  =    6\u221a3                                                                             (\u221a3 + \u221a2) (\u221a3 &#8211; \u221a2)<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week9SS1Fi6.png\" alt=\"\"\/>         3                                                                              =       3\u221a3 &#8211; 3\u221a2<br \/>\n  = 2\u221a3                                                                                     (\u221a3)<sup>2<\/sup> \u2013 (\u221a2)<sup>2<\/sup><br \/>\n\t\t                                                                                         =         3\u221a3 &#8211; 3\u221a2<strong><br \/>\n\t\t\t\t<\/strong>                                                                                                       3 &#8211; 1<br \/>\n<strong>                                                                   \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0     <\/strong>=     3\u221a3 &#8211; 3\u221a2<br \/>\n\t\t                                                                           \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0  1<br \/>\n                                                                                         =      3(\u221a3 -\u221a2)<br \/>\n<strong>Equality of Surds<br \/>\n<\/strong>Given two surds i.e  P + \u221am   and q + \u221an     if P +\u221am = q + \u221an    then<br \/>\n\u221aP &#8211; q   = \u221an &#8211;  m   the L.H.S<br \/>\nOf the equation is a rational number while the L.H.S and R.H.S can only be equal if they are both equal to zero (0)<br \/>\n\u00a0\u00a0\u00a0\u00a0   P \u2013 q = 0<br \/>\n\u00a0\u00a0\u00a0\u00a0:. P = q and n &#8211; m = 0 i.e.<br \/>\n            \u221an  = \u221am<\/p>\n<p>\u00a0<strong>Examples<\/strong>:<br \/>\n Find the square root of the following?<br \/>\na)\u00a0\u00a0\u00a0\u00a07 + 2\u221a10\u00a0\u00a0\u00a0\u00a0b)  14 &#8211; 4\u221a6<\/p>\n<p>\u00a0<strong>Solution<br \/>\n<\/strong>(a) Let the square root of 7 + 2 \u221a10 be \u221am + \u221an<br \/>\n(\u221am + \u221an)<sup>2<\/sup> = 7  + 2\u221a10<br \/>\nm +\u221a2mn+ n = 7 + 2\u221a10<br \/>\n m + n   = 7                                 (1)<br \/>\n2\u221amn   = 2\u221a10<br \/>\n    mn   =   10<br \/>\nSquaring both surds we have<br \/>\nmn =  10  _______(ii)<br \/>\nm + n = 7 ______ (i)<br \/>\nm n = 10   _______ (ii)<br \/>\n        From equation (1)    m = 7 \u2013 n<br \/>\n        Put m in (ii)   we have<br \/>\n\u00a0\u00a0\u00a0\u00a0(7 \u2013 n) n = 10<br \/>\n7n \u2013 n<sup>2<\/sup> = 10<br \/>\n        In sum; n<sup>2<\/sup> \u2013 7n + 10 = 0<br \/>\n\u00a0\u00a0\u00a0\u00a0n<sup>2<\/sup> \u2013 2n \u2013 5n + 10 =0<br \/>\n \u00a0\u00a0\u00a0\u00a0n (n \u2013 2) \u2013 5 (n \u2013 2) = 0<br \/>\n\u00a0\u00a0\u00a0\u00a0(n -5) (n \u2013 2) = 0<br \/>\nn = 5 or 2<br \/>\n            m = 7 \u2013 2, where n = 2<br \/>\n\u00a0\u00a0\u00a0\u00a0m = 5,<br \/>\n            m = 7 \u2013 5 , when n = 5<br \/>\n\u00a0\u00a0\u00a0\u00a0m = 2<br \/>\n\u00a0\u00a0\u00a0\u00a0m= 5 or 2<br \/>\n          The square root of 7 + \u221a10 are 5 &amp; + 2.<\/p>\n<p>\u00a0(b) Let the square root of 14 \u2013 4\u221a6   be \u221aP &#8211; \u221aQ<br \/>\n           The (\u221aP &#8211; \u221aQ) <sup>2 <\/sup> =14 \u2013 4\u221a6\u00a0\u00a0\u00a0\u00a0<br \/>\nP &#8211; 2\u221aPQ + Q = 14 \u2013 4 \u221a6<br \/>\n            P + Q =   14 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 (1)<\/p>\n<p>\u00a0          -2\u221aPQ      =  &#8211; 4\u221a6<br \/>\n<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week9SS1Fi7.png\" alt=\"\"\/><img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0727_Week9SS1Fi8.png\" alt=\"\"\/>              -2               &#8211; 2<br \/>\n          \u221aPQ =   2\u221a6      (squaring both sides)<\/p>\n<p>\u00a0           PQ = (2\u221a6)<sup>2<\/sup><br \/>\n\t\t           PQ = 4 x 6 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. (11)<\/p>\n<p>\u00a0           P + Q = 14 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 (1)<\/p>\n<p>\u00a0           PQ = 24 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 (11)<br \/>\n          From equation\u2026\u2026\u2026\u2026\u2026\u2026 (1)   P = 14 &#8211; Q<br \/>\n          Sub for p in equation \u2026\u2026\u2026\u2026\u2026\u2026\u2026 (11)<br \/>\n          (14 \u2013 Q) Q = 24<br \/>\n           14Q \u2013 Q<sup>2<\/sup> = 24<br \/>\n          In turn we have:<br \/>\n          Q<sup>2<\/sup> \u2013 14Q + 24 = 0<br \/>\n          Q<sup>2<\/sup> \u2013 12Q \u2013 2Q + 24 = 0<br \/>\n          Q (Q -12) \u2013 2 (Q \u2013 12) = 0<br \/>\n          Q = 2 or 12<br \/>\n         If  P = 14 \u2013 Q ,when Q= 12<br \/>\n         P = 14 \u2013 12<br \/>\n         P = 2<br \/>\n         If P = 14 \u2013 Q when Q = 2<br \/>\n         P = 14 &#8211; 2<br \/>\n             = 12<br \/>\n       (\u221a12  \u2013   \u221a2 )\u00a0\u00a0\u00a0\u00a0= (2 \u221a 3 &#8211; \u221a2)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0and<br \/>\n       (\u221a2   &#8211; \u221a12) \u00a0\u00a0\u00a0\u00a0=  (\u221a2 &#8211;\u00a0\u00a0\u00a0\u00a0 2 \u221a3)<\/p>\n<p>\u00a0<strong>Evaluation:<br \/>\n<\/strong>1. Express  3\u221a2 &#8211; \u221a3 in the form \u221am where m and n are whole number.<br \/>\n                  2\u221a3 &#8211; \u221a2                    \u221an<\/p>\n<p>\u00a02. Express          1  <\/p>\n<p>\t\t\tin the form p\u221a5 + q\u221a3, where p and q are rational numbers.<strong><br \/>\n\t\t\t<\/strong>                     \u221a5 +\u221a3<\/p>\n<p>\u00a0<strong>General Evaluation<br \/>\n<\/strong>1. Simplify      3x<sup>2<\/sup> x 4x<sup>3<\/sup><br \/>\n\t\t\t                            6x<sup>7<\/sup><br \/>\n\t\t2. Evaluate   23.97   x   0.7124<br \/>\n\t\t         3.877   x   52.18<br \/>\n3. Solve 9<sup>(1 &#8211; x)<\/sup> = (1\/27) <sup>x+1<\/sup><br \/>\n\t\t4. Log<sub>8<\/sub> (r<sup>2<\/sup> \u2013 8r + 18) = 1\/3<br \/>\n5. Simplify: 2\u221a12 + 3\u221a48 + \u221a75<\/p>\n<p>\u00a0<strong>Reading Assignment<\/strong>: Further Mathematics Project Book 1(New third edition).Chapter 3 pg.19-27<\/p>\n<p>\u00a0<strong>Weekend Assignment<br \/>\n<\/strong>1. Expand (3\u221a2 &#8211; 1) (3\u221a2 + 1)                      (a) 16         (b) 20      (c) 17           (d) 24<br \/>\n2. Simplify \u221a200 in its basic form                (a) 10\u221a2     (b) 5\u221a4    (c) 2\u221a10       (d) 2\u221a50<br \/>\n3. Simplify 9\/\u221a3                                            (a) 3\u221a2       (b) 3\u221a3    (c) 1\/3          (d) 2\u221a2<br \/>\n4. Express 3\u221a5 as a single surd                     (a) \u221a40        (b) \u221a55    (c) \u221a45         (d) \u221a35implify<br \/>\n5. Simplify \u221a`128 &#8211; 4\u221a8                                 (a) 0            (b) 1        (c) 2             (d) 3<br \/>\n<strong>Theory<br \/>\n<\/strong>1.Express  3\u221a2 &#8211; \u221a3   in the form \u221am where m and n are whole number.<br \/>\n                   2\u221a3 &#8211; \u221a2                     \u221an<br \/>\n2.Express          1   in the form <strong><em>p\u221a5 + q\u221a3<\/em><\/strong>, where p and q are rational numbers.<br \/>\n                   \u221a5 +\u221a3<\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<strong><br \/>\n\t\t\t<\/strong>\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u00a0 \u00a0WEEK NINE TOPIC: SURDS CONTENT Rules of surds Basic Form of Surds Similar Surds&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,173],"tags":[],"class_list":["post-1933","post","type-post","status-publish","format-standard","hentry","category-posts","category-first-term-ss1-further-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/1933","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/comments?post=1933"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/1933\/revisions"}],"predecessor-version":[{"id":1934,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/1933\/revisions\/1934"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=1933"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=1933"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=1933"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}