{"id":118,"date":"2022-04-12T12:09:38","date_gmt":"2022-04-12T12:09:38","guid":{"rendered":"http:\/\/8thclass.deltapublications.in\/?page_id=118"},"modified":"2025-10-14T09:35:42","modified_gmt":"2025-10-14T09:35:42","slug":"f-15-positive-and-negative-square-roots","status":"publish","type":"page","link":"https:\/\/8thclass.deltapublications.in\/index.php\/f-15-positive-and-negative-square-roots\/","title":{"rendered":"F.15 Positive and negative square roots"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color has-huge-font-size\" style=\"color:#00056d;text-transform:uppercase\"><strong>Positive and negative square roots<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-09eb8d27d0d70f35bd20b1cf47293f11\" style=\"color:#74008b\"><strong>Key Notes :<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading has-primary-color has-text-color has-link-color wp-elements-9d82b40c2a0b1b21d01f73f83572e162\">\ud83e\uddee<strong> Positive and Negative Square Roots<\/strong><\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\ud83c\udf1f <strong>What is a Square Root?<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The <strong>square root<\/strong> of a number is a value that, when <strong>multiplied by itself<\/strong>, gives the <strong>original number<\/strong>.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\ud83d\udc49 <strong>Example<\/strong>: \u221a9 = 3 because 3 \u00d7 3 = 9<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\u2795 <strong>Positive Square Root<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The <strong>positive square root<\/strong> is the <strong>non-negative<\/strong> value of a number\u2019s square root.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2705 Example: \u221a25 = <strong>+5<\/strong><\/li>\n<\/ul>\n\n\n\n<p>\ud83d\udcd8 We call it the <strong>principal square root<\/strong>.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The symbol <strong>\u221a<\/strong> always represents the <strong>positive root<\/strong> unless stated otherwise.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\u2796 <strong>Negative Square Root<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Every <strong>positive number<\/strong> also has a <strong>negative square root<\/strong>, because a <strong>negative times a negative<\/strong> equals a <strong>positive<\/strong>.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\ud83d\udc49 <strong>Example<\/strong>: \u22125 \u00d7 \u22125 = 25 \u2192 So, \u221a25 = <strong>\u00b15<\/strong><\/li>\n<\/ul>\n\n\n\n<p>\ud83d\udca1 Therefore, <strong>a positive number has two square roots<\/strong>: <\/p>\n\n\n\n<p class=\"has-text-align-center\">\u221aa=+number&nbsp;and&nbsp;\u2212number<\/p>\n\n\n\n<p>Example:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u221a36 = <strong>\u00b16<\/strong> \u2192 (that means +6 and \u22126)<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\ud83d\udeab <strong>Square Roots of Negative Numbers<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The square root of a <strong>negative number<\/strong> is <strong>not a real number<\/strong>.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2716 <strong>Example<\/strong>: \u221a(\u22129) = <strong>no real value<\/strong>, because no real number squared gives a negative result.<\/li>\n\n\n\n<li>(It belongs to <strong>imaginary numbers<\/strong>, represented using <em>i<\/em> in higher grades.)<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\ud83d\udcd0 <strong>Perfect Squares and Their Roots<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Number<\/th><th>Positive \u221a<\/th><th>Negative \u221a<\/th><\/tr><\/thead><tbody><tr><td>4<\/td><td>+2<\/td><td>\u22122<\/td><\/tr><tr><td>9<\/td><td>+3<\/td><td>\u22123<\/td><\/tr><tr><td>16<\/td><td>+4<\/td><td>\u22124<\/td><\/tr><tr><td>25<\/td><td>+5<\/td><td>\u22125<\/td><\/tr><tr><td>36<\/td><td>+6<\/td><td>\u22126<\/td><\/tr><tr><td>49<\/td><td>+7<\/td><td>\u22127<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\ud83e\udde0 <strong>Key Points to Remember<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>\u2705 Every <strong>positive number<\/strong> has <strong>two square roots<\/strong> (one positive, one negative).<\/p>\n\n\n\n<p>\u2705 <strong>0<\/strong> has <strong>only one square root<\/strong>, which is <strong>0<\/strong>.<\/p>\n\n\n\n<p>\u2705 <strong>Negative numbers<\/strong> do <strong>not<\/strong> have real square roots.<\/p>\n\n\n\n<p>\u2705 The <strong>principal (positive)<\/strong> square root is written with the <strong>\u221a symbol<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\ud83c\udf08 <strong>Examples<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>1\ufe0f\u20e3 \u221a49 = \u00b17<\/p>\n\n\n\n<p>2\ufe0f\u20e3 \u221a64 = \u00b18<\/p>\n\n\n\n<p>3\ufe0f\u20e3 \u221a0 = 0<\/p>\n\n\n\n<p>4\ufe0f\u20e3 \u221a(\u221216) \u2192 <strong>Not a real number<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\ud83d\udcac <strong>Shortcut Tip<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li>If x<sup>2<\/sup>=25, then x=\u00b15<\/li>\n\n\n\n<li>\ud83d\udc49 Always remember to write <strong>both roots<\/strong> when solving equations like this!<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-text-color has-link-color has-large-font-size wp-elements-3fefd20f8ce119a518e5f53333ab5486\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group has-primary-color has-text-color has-background has-link-color has-normal-font-size wp-elements-0b9d50f60c424f4fb8c3a00a7a027b92\" style=\"background-color:#c1f3f4\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-background-background-color has-text-color has-background\" style=\"color:#b00012\"><strong>What&nbsp;is the negative square root of&nbsp;16?<\/strong><\/p>\n\n\n\n<p>Start&nbsp;by finding the positive square root of&nbsp;16.<\/p>\n\n\n\n<p>Figure&nbsp;out which number squared (multiplied by itself) equals&nbsp;16.<\/p>\n\n\n\n<p>The&nbsp;number&nbsp;4&nbsp;squared equals&nbsp;16.<\/p>\n\n\n\n<p class=\"has-text-align-center\">4<sup>2<\/sup>= 4.4 =16<\/p>\n\n\n\n<p>So&nbsp;the positive square root of&nbsp;16&nbsp;is&nbsp;4. <\/p>\n\n\n\n<p>Finally,&nbsp;include the negative sign to get the negative square&nbsp;root.<\/p>\n\n\n\n<p>The&nbsp;negative square root of&nbsp;16&nbsp;is&nbsp;\u20134.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-primary-color has-text-color has-background has-link-color has-normal-font-size wp-elements-1b8aeb2b81b09b5a9e5e9b9b0d3046ce\" style=\"background-color:#feedb3\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-background-background-color has-text-color has-background\" style=\"color:#b00012\"><strong>What&nbsp;is&nbsp;\u2013\u221a9?<\/strong><\/p>\n\n\n\n<p>Start&nbsp;by finding the positive square root of&nbsp;9. <\/p>\n\n\n\n<p>Figure&nbsp;out which number squared (multiplied by itself) equals&nbsp;9.<\/p>\n\n\n\n<p>The&nbsp;number&nbsp;3&nbsp;squared equals&nbsp;9.<\/p>\n\n\n\n<p>3<sup>2<\/sup>= 3.3 =9<\/p>\n\n\n\n<p>So&nbsp;the positive square root of&nbsp;9&nbsp;is&nbsp;3.<\/p>\n\n\n\n<p>Finally,&nbsp;include the negative sign to get the negative square&nbsp;root.<\/p>\n\n\n\n<p>\u2013\u221a9&nbsp;is&nbsp;\u20133.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-primary-color has-text-color has-background has-link-color has-normal-font-size wp-elements-35a68677da8a44654d02af86fc3906df\" style=\"background-color:#f2bcbc\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"has-background-background-color has-text-color has-background has-link-color wp-elements-cd1bc353cbd30f3dcf363ea3ffe8b5fd\" style=\"color:#b00012\"><strong>\ud83d\udd25What&nbsp;is the negative square root of&nbsp;81?<\/strong><\/p>\n\n\n\n<p>Start&nbsp;by finding the positive square root of&nbsp;81.<\/p>\n\n\n\n<p>Figure&nbsp;out which number squared (multiplied by itself) equals&nbsp;81.<\/p>\n\n\n\n<p>The&nbsp;number&nbsp;9&nbsp;squared equals&nbsp;81.<\/p>\n\n\n\n<p>9<sup>2<\/sup>=9 . 9=81<\/p>\n\n\n\n<p>So&nbsp;the positive square root of&nbsp;81&nbsp;is&nbsp;9.<\/p>\n\n\n\n<p>Finally,&nbsp;include the negative sign to get the negative square&nbsp;root.<\/p>\n\n\n\n<p>The&nbsp;negative square root of&nbsp;81&nbsp;is&nbsp;\u20139.<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\"><strong>Let&#8217;s try some examples! \u270d\ufe0f<\/strong><\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/78202\/199\/744\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-311.png\" alt=\"\" class=\"wp-image-9636\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-311.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-311-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-311-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/78340\/701\/371\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-318.png\" alt=\"\" class=\"wp-image-9637\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-318.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-318-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-318-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Positive and negative square roots Key Notes : \ud83e\uddee Positive and Negative Square Roots \ud83c\udf1f What is a Square Root? The square root of a number is a value that, when multiplied by itself, gives the original number. \u2795 Positive Square Root The positive square root is the non-negative value of a number\u2019s square root.<a class=\"more-link\" href=\"https:\/\/8thclass.deltapublications.in\/index.php\/f-15-positive-and-negative-square-roots\/\">Continue reading <span class=\"screen-reader-text\">&#8220;F.15 Positive and negative square roots&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-118","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_hostinger_reach_plugin_has_subscription_block":false,"_hostinger_reach_plugin_is_elementor":false,"_links":{"self":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/118","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=118"}],"version-history":[{"count":28,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/118\/revisions"}],"predecessor-version":[{"id":21882,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/118\/revisions\/21882"}],"wp:attachment":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}