{"id":27847,"date":"2022-01-25T17:12:34","date_gmt":"2022-01-25T11:42:34","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=27847"},"modified":"2022-01-25T17:12:34","modified_gmt":"2022-01-25T11:42:34","slug":"squares-and-square-roots-class-8-notes-maths-chapter-6","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/study-material\/squares-and-square-roots\/class-8\/notes\/maths\/chapter-6\/","title":{"rendered":"Squares and Square Roots Class 8 Notes Maths Chapter 6"},"content":{"rendered":"<p>&nbsp;<\/p>\n<h2><a href=\"https:\/\/infinitylearn.com\/surge\/wp-admin\/post.php?post=25759&amp;action=edit\">CBSE Class 8 Maths Notes Chapter 6 Squares and Square Roots<\/a><\/h2>\n<p><strong>Square Number:<\/strong><br \/>\nThe square of a number is the product of the number with the number itself Thus, square of x = (x \u00d7 x), denoted by x<sup>2<\/sup>.<br \/>\nA natural number n is a perfect square, if n = m<sup>2<\/sup> for some natural number m.{1 = 1 \u00d7 1 = 1<sup>2<\/sup>, 4 = 2 \u00d7 2 = 2<sup>2<\/sup>}<\/p>\n<p><strong>Square Root:<\/strong><br \/>\nSquare root is the inverse operation of square, i.e., positive square root of a number is denoted by the symbol \u221a<br \/>\nFor example, 3<sup>2<\/sup> = 9 gives \u221a9 = 3 or (3<sup>2<\/sup>)<sup>1\/2<\/sup> = 3.<br \/>\nFor positive numbers a and b, we have<br \/>\n<img loading=\"lazy\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2021\/12\/46812777725_ef2144bd56_o.png\" alt=\"Squares and Square Roots Class 8 Notes Maths Chapter 6 1\" width=\"482\" height=\"73\" \/><\/p>\n<p><strong>Properties of the Square Number:<\/strong><\/p>\n<p>A number ending in 2, 3, 7 or 8 is never a perfect square.<\/p>\n<p>A number ending in an odd number of zeros is never a perfect square.<\/p>\n<p>The square of an even number is even.<\/p>\n<p>The square of an odd number is odd.<\/p>\n<p>The square of a proper fraction is smaller than the fraction.<\/p>\n<p>For every natural number n, we have {(n + 1)<sup>2<\/sup> \u2013 n<sup>2<\/sup>} = {(n + 1) + n}.<\/p>\n<p>Sum of first n odd natural numbers = n<sup>2<\/sup>.<\/p>\n<p>If m, n, p are natural numbers such that (m<sup>2<\/sup> + n<sup>2<\/sup>) = p<sup>2<\/sup>, then (m, n, p) is called a Pythagorean triplet.<\/p>\n<p>For every natural number m &gt; 1, (2m, m<sup>2<\/sup> \u2013 1, m<sup>2<\/sup> + 1) is a Pythagorean triplet.<\/p>\n<p>There are 2n non-perfect square numbers between the squares of the number n and (n + 1).<\/p>\n<p>The numbers which can be expressed as the product of the number with itself are called square numbers or perfect squares.<br \/>\nFor example, 1, 4, 9, 16, 25, \u2026.<\/p>\n<p>If a natural number m can be expressed as n<sup>2<\/sup>, where n is also a natural number, then, m is called a square number.<\/p>\n<p><strong>Some More Interesting Patterns<br \/>\n<\/strong>A triangular number is one whose dot patterns can be arranged as triangles. If we combine two consecutive triangular numbers, we get a square number.<\/p>\n<p>There are 2n non-perfect square numbers between the squares of the numbers n and (n + 1) which is 1 less than the difference of two squares.<\/p>\n<p>Sum of first n odd natural numbers is n<sup>2<\/sup>.<\/p>\n<p>If a natural number cannot be expressed as a sum of successive odd natural number starting with 1, then it is not a perfect square. In other words, if a natural number is a square number, then it is necessarily the sum of successive odd numbers starting with 1. This result can be used to decide whether a given natural number is a perfect square or not.<\/p>\n<p>We can express the square of any odd number as the sum of two consecutive positive integers.<br \/>\n<img loading=\"lazy\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2021\/12\/40762765293_a0c1b3af20_o.png\" alt=\"Squares and Square Roots Class 8 Notes Maths Chapter 6 2\" width=\"642\" height=\"60\" \/><\/p>\n<p>If (n + 1) and (n \u2013 1) are two consecutive even or odd natural numbers, then (n + 1) \u00d7 (n \u2013 1) = n<sup>2<\/sup> \u2013 1.<\/p>\n<p><strong>Finding the Square of a Number<\/strong><br \/>\nSee 23<sup>2<\/sup> = (20 + 3)<sup>2<\/sup> = (20 + 3) (20 + 3)<br \/>\n= 20 (20 + 3) + 3 (20 + 3)<br \/>\n= 20<sup>2<\/sup> + 20 \u00d7 3 + 3 \u00d7 20 + 3<sup>2<\/sup><br \/>\n= 400 + 60 + 60 + 9<br \/>\n= 529<\/p>\n<p><strong>Other Patterns in Squares<\/strong><br \/>\nLet a 5 be a number with unit digit 5. Then, (a5)<sup>2<\/sup> = a (a + 1) hundred +25<\/p>\n<p><strong>Pythagorean Triplets Important Points<br \/>\n<\/strong>If a, b and c are three numbers such that any one of the following three relations holds:<br \/>\n(i) a<sup>2<\/sup> + b<sup>2<\/sup> = c<sup>2<\/sup><br \/>\n(ii) b<sup>2<\/sup> + c<sup>2<\/sup> = a<sup>2<\/sup><br \/>\n(iii) c<sup>2<\/sup> + a<sup>2<\/sup> = b<sup>2<\/sup><br \/>\nthen the numbers a, b, c are said to form a Pythagorean triplet.<br \/>\nFor example: 3, 4, 5 is a Pythagorean triplet because 3<sup>2<\/sup> + 4<sup>2<\/sup> = 9 + 16 = 25 = 5<sup>2<\/sup>.<br \/>\nwe can find more such triplets.<br \/>\nFor example: 8, 15, 17; 12, 9, 15; 12, 35, 37 etc.<\/p>\n<p><strong>Square Roots<\/strong><br \/>\nThe square root of a number \u2018a\u2019 is that number which when multiplied by itself gives that number \u2018a\u2019 as product.<br \/>\nThus, if b is the square root of a;<br \/>\nthen b \u00d7 b = a or b<sup>2<\/sup> = a<br \/>\nSymbolically, we write b = \u221aa<br \/>\nNote: b= \u221aa \u21d4 b<sup>2<\/sup> = a<br \/>\ni.e., b is the square root of an if and only if a is the square of b.<\/p>\n<p><strong>Finding Square Roots<\/strong><br \/>\nTo find, a number whose square is known is known as finding the square root.<br \/>\nFinding the square root is the inverse (opposite) operation of squaring.<br \/>\nThere are two integral square roots of a perfect square number.<br \/>\nFor example: 4 = (2)<sup>2<\/sup> = (-2)<sup>2<\/sup><br \/>\n\u221a4 = 2 and 2 both. Here, we shall take up only positive square root of a natural number.<br \/>\nThus, \u221a4 = 2 (not -2)<br \/>\nThe positive square root of a number is denoted by the symbol \u221a.<br \/>\nFor example, 3<sup>2<\/sup> = 9 \u21d2 \u221a9 = 3<\/p>\n<p><strong>Finding Square Root Through Repeated Subtraction<\/strong><br \/>\nWe subtract successive odd number starting from 1 from the given square number till we get zero. The number of times, we have to make subtractions, is called the square root of the given square number.<\/p>\n<p><strong>Finding Square Root Through Prime Factorisation<\/strong><br \/>\nWe find the prime factors of the given perfect square and arrange in pairs. Then, we choose one factor from each pair and multiply together. The product thus obtained gives the required square root.<br \/>\nNote: A square number has complete pairs of its prime factors.<\/p>\n<p><strong>Finding Square Root By Division Method<\/strong><br \/>\n<strong>Steps<\/strong><br \/>\n(i) Obtain the number whose square root is to be found. Place a bar over every pair of digits starting from the digit at one\u2019s place. If the number of digits in it is odd, then the left-most single digit too will have a bar. Each pair and the left-most single digit (if any) is called a period.<\/p>\n<p>(ii) Think of the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend. Divide and get the remainder.<\/p>\n<p>(iii) Bring down the number under the next bar to the right of the remainder. This becomes the new dividend.<\/p>\n<p>(iv) Double the divisor and enter it with a blank on its right.<\/p>\n<p>(v) Guess a largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient, the product is less than or equal to the new dividend obtained in step (iii).<\/p>\n<p>(vi) Continue this process till the remainder is 0 and no digits are left in the given number. The quotient thus obtained is the required square root of the given number.<\/p>\n<p><strong>Square Roots of Decimals<\/strong><br \/>\nPut bars on an integral part of the number in the usual manner. Place bars in the decimal part on every pair of digits beginning with the first decimal place. Proceed as usual to find the square root.<\/p>\n<p>We hope the given CBSE Class 8 Maths Notes Chapter 6 Squares and Square Roots Pdf free download will help you. If you have any query regarding NCERT Class 8 Maths Notes Chapter 6 Squares and Square Roots, drop a comment below and we will get back to you at the earliest.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>&nbsp; CBSE Class 8 Maths Notes Chapter 6 Squares and Square Roots Square Number: The square of a number is [&hellip;]<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Get CBSE Class 8 Maths Notes Chapter 6 Squares and Square Roots to infinity learn.","custom_permalink":"study-material\/squares-and-square-roots\/class-8\/notes\/maths\/chapter-6\/"},"categories":[92,21],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Squares and Square Roots Class 8 Notes Maths Chapter 6 - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"Get CBSE Class 8 Maths Notes Chapter 6 Squares and Square Roots to infinity learn.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/study-material\/squares-and-square-roots\/class-8\/notes\/maths\/chapter-6\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Squares and Square Roots Class 8 Notes Maths Chapter 6 - 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