{"id":8656,"date":"2023-10-04T15:40:09","date_gmt":"2023-10-04T15:40:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8656"},"modified":"2024-10-18T20:56:44","modified_gmt":"2024-10-18T20:56:44","slug":"irrational-numbers-learn-it-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/irrational-numbers-learn-it-3\/","title":{"raw":"Irrational Numbers: Learn It 3","rendered":"Irrational Numbers: Learn It 3"},"content":{"raw":"<h2>Simplifying Square Roots and Expressing Them in Lowest Terms<\/h2>\r\n<p>To <strong>simplify a square root<\/strong> means that we rewrite the square root as a rational number times the square root of a number that has no perfect square factors. The act of changing a square root into such a form is simplifying the square root.<\/p>\r\n<section class=\"textbox recall\">\r\n<p>The number inside the square root symbol is referred to as the radicand. So in the expression [latex]\\sqrt{a}[\/latex] the number [latex]a[\/latex] is referred to as the radicand.<\/p>\r\n<\/section>\r\n<p>Before discussing how to simplify a square root, we need to introduce a rule about square roots.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>the product rule for square roots<\/h3>\r\n<p>The square root of a product of numbers equals the product of the square roots of those number.<\/p>\r\n<p>Given that [latex]a[\/latex] and [latex]b[\/latex] are nonnegative real numbers, <center>[latex]\\sqrt{a \\times {b}}=\\sqrt{a} \\times \\sqrt{b}[\/latex]<\/center><\/p>\r\n<\/div>\r\n<\/section>\r\n<p>Using this formula, we can factor an integer inside a square root into a perfect square times another integer. Then the square root can be applied to the perfect square, leaving an integer times the square root of another integer. If the number remaining under the square root has no perfect square factors, then we\u2019ve simplified the irrational number into lowest terms.<\/p>\r\n<section class=\"textbox proTip\">\r\n<p>A perfect square is an integer that can be expressed as the square of another integer. For example, [latex]16[\/latex], [latex]25[\/latex], and [latex]36[\/latex] are perfect squares because they are [latex]4^2[\/latex], [latex]5^2[\/latex], and [latex]6^2[\/latex], respectively.<\/p>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How to: To simplify the irrational number into lowest terms when [latex]n[\/latex] is an integer<\/strong><\/p>\r\n<ul>\r\n\t<li><strong>Step 1:<\/strong> Determine the largest perfect square factor of [latex]n[\/latex], which we denote [latex]a^2[\/latex].<\/li>\r\n\t<li><strong>Step 2:<\/strong> Factor [latex]n[\/latex] into [latex]a^2\u00d7b[\/latex].<\/li>\r\n\t<li><strong>Step 3:<\/strong> Apply [latex]\\sqrt{a^2 \\times b} =\\sqrt{a^2} \\times \\sqrt{b}[\/latex].<\/li>\r\n\t<li><strong>Step 4:<\/strong> Write [latex]\\sqrt{n}[\/latex] in its simplified form, [latex]a\\sqrt{b}[\/latex].<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>When a square root has been simplified in this manner, [latex]a[\/latex] is referred to as the rational part of the number, and [latex]\\sqrt{b}[\/latex] is referred to as the irrational part.<\/p>\r\n<section class=\"textbox example\">\r\n<p>Simplify the irrational number [latex]\\sqrt{180}[\/latex] and express in lowest terms. Identify the rational and irrational parts.<\/p>\r\n<p>[reveal-answer q=\"214538\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214538\"]<\/p>\r\n<p>Begin by finding the largest perfect square that is a factor of [latex]180[\/latex]. We can do this by writing out the factor pairs of [latex]180[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]1 \\times 180, \\enspace 2 \\times 90, \\enspace 3 \\times 60, \\enspace 4 \\times 45, \\enspace 5 \\times 36, \\enspace 6 \\times 30, \\enspace 9 \\times 20, \\enspace 10 \\times 18, \\enspace 12 \\times 15[\/latex]<\/p>\r\n<p>Looking at the list of factors, the perfect squares are [latex]4[\/latex], [latex]9[\/latex], and [latex]36[\/latex]. The largest is [latex]36[\/latex], so we factor the into [latex]36\u00d75=6^2\u00d75[\/latex]. In the formula, [latex]a=6[\/latex] and [latex]b=5[\/latex].<\/p>\r\n<p>Apply [latex]\\sqrt{a^2 \\times b}=\\sqrt{a^2} \\times \\sqrt{b}[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\sqrt{6^2 \\times 5}=\\sqrt{6^2} \\times \\sqrt{5}[\/latex]<\/p>\r\n<p>The simplified form of [latex]\\sqrt{180}[\/latex] is [latex]6\\sqrt{5}[\/latex]. In this example, the [latex]6[\/latex] is the rational part, and the [latex]\\sqrt{5}[\/latex] is the irrational part.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Simplify the irrational number [latex]\\sqrt{330}[\/latex] and express in lowest terms. Identify the rational and irrational parts.<\/p>\r\n<p>[reveal-answer q=\"214558\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214558\"]<\/p>\r\n<p>Begin by finding the largest perfect square that is a factor of [latex]330[\/latex]. We can do this by writing out the factor pairs of [latex]330[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]1 \\times 330, \\enspace  2 \\times 165, \\enspace 3 \\times 110, \\enspace 5 \\times 66, \\enspace 6 \\times 55, \\enspace 10 \\times 33, \\enspace 11 \\times 30, \\enspace 15 \\times 22[\/latex]<\/p>\r\n<p>Looking at the list of factors, there are no perfect squares other than [latex]1[\/latex], which means [latex]\\sqrt{330}[\/latex] is already expressed in lowest terms.<\/p>\r\n<p>In this case, [latex]1[\/latex] is the rational part, and [latex]\\sqrt{330}[\/latex] is the irrational part. Though we could write this as [latex]1\\sqrt{330}[\/latex], but the product of [latex]1[\/latex] and any other number is just the number.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]12704[\/ohm2_question]<\/p>\r\n<\/section>","rendered":"<h2>Simplifying Square Roots and Expressing Them in Lowest Terms<\/h2>\n<p>To <strong>simplify a square root<\/strong> means that we rewrite the square root as a rational number times the square root of a number that has no perfect square factors. The act of changing a square root into such a form is simplifying the square root.<\/p>\n<section class=\"textbox recall\">\n<p>The number inside the square root symbol is referred to as the radicand. So in the expression [latex]\\sqrt{a}[\/latex] the number [latex]a[\/latex] is referred to as the radicand.<\/p>\n<\/section>\n<p>Before discussing how to simplify a square root, we need to introduce a rule about square roots.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the product rule for square roots<\/h3>\n<p>The square root of a product of numbers equals the product of the square roots of those number.<\/p>\n<p>Given that [latex]a[\/latex] and [latex]b[\/latex] are nonnegative real numbers, <\/p>\n<div style=\"text-align: center;\">[latex]\\sqrt{a \\times {b}}=\\sqrt{a} \\times \\sqrt{b}[\/latex]<\/div>\n<\/div>\n<\/section>\n<p>Using this formula, we can factor an integer inside a square root into a perfect square times another integer. Then the square root can be applied to the perfect square, leaving an integer times the square root of another integer. If the number remaining under the square root has no perfect square factors, then we\u2019ve simplified the irrational number into lowest terms.<\/p>\n<section class=\"textbox proTip\">\n<p>A perfect square is an integer that can be expressed as the square of another integer. For example, [latex]16[\/latex], [latex]25[\/latex], and [latex]36[\/latex] are perfect squares because they are [latex]4^2[\/latex], [latex]5^2[\/latex], and [latex]6^2[\/latex], respectively.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: To simplify the irrational number into lowest terms when [latex]n[\/latex] is an integer<\/strong><\/p>\n<ul>\n<li><strong>Step 1:<\/strong> Determine the largest perfect square factor of [latex]n[\/latex], which we denote [latex]a^2[\/latex].<\/li>\n<li><strong>Step 2:<\/strong> Factor [latex]n[\/latex] into [latex]a^2\u00d7b[\/latex].<\/li>\n<li><strong>Step 3:<\/strong> Apply [latex]\\sqrt{a^2 \\times b} =\\sqrt{a^2} \\times \\sqrt{b}[\/latex].<\/li>\n<li><strong>Step 4:<\/strong> Write [latex]\\sqrt{n}[\/latex] in its simplified form, [latex]a\\sqrt{b}[\/latex].<\/li>\n<\/ul>\n<\/section>\n<p>When a square root has been simplified in this manner, [latex]a[\/latex] is referred to as the rational part of the number, and [latex]\\sqrt{b}[\/latex] is referred to as the irrational part.<\/p>\n<section class=\"textbox example\">\n<p>Simplify the irrational number [latex]\\sqrt{180}[\/latex] and express in lowest terms. Identify the rational and irrational parts.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214538\">Show Solution<\/button> <\/p>\n<div id=\"q214538\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by finding the largest perfect square that is a factor of [latex]180[\/latex]. We can do this by writing out the factor pairs of [latex]180[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]1 \\times 180, \\enspace 2 \\times 90, \\enspace 3 \\times 60, \\enspace 4 \\times 45, \\enspace 5 \\times 36, \\enspace 6 \\times 30, \\enspace 9 \\times 20, \\enspace 10 \\times 18, \\enspace 12 \\times 15[\/latex]<\/p>\n<p>Looking at the list of factors, the perfect squares are [latex]4[\/latex], [latex]9[\/latex], and [latex]36[\/latex]. The largest is [latex]36[\/latex], so we factor the into [latex]36\u00d75=6^2\u00d75[\/latex]. In the formula, [latex]a=6[\/latex] and [latex]b=5[\/latex].<\/p>\n<p>Apply [latex]\\sqrt{a^2 \\times b}=\\sqrt{a^2} \\times \\sqrt{b}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{6^2 \\times 5}=\\sqrt{6^2} \\times \\sqrt{5}[\/latex]<\/p>\n<p>The simplified form of [latex]\\sqrt{180}[\/latex] is [latex]6\\sqrt{5}[\/latex]. In this example, the [latex]6[\/latex] is the rational part, and the [latex]\\sqrt{5}[\/latex] is the irrational part.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Simplify the irrational number [latex]\\sqrt{330}[\/latex] and express in lowest terms. Identify the rational and irrational parts.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214558\">Show Solution<\/button> <\/p>\n<div id=\"q214558\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by finding the largest perfect square that is a factor of [latex]330[\/latex]. We can do this by writing out the factor pairs of [latex]330[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]1 \\times 330, \\enspace  2 \\times 165, \\enspace 3 \\times 110, \\enspace 5 \\times 66, \\enspace 6 \\times 55, \\enspace 10 \\times 33, \\enspace 11 \\times 30, \\enspace 15 \\times 22[\/latex]<\/p>\n<p>Looking at the list of factors, there are no perfect squares other than [latex]1[\/latex], which means [latex]\\sqrt{330}[\/latex] is already expressed in lowest terms.<\/p>\n<p>In this case, [latex]1[\/latex] is the rational part, and [latex]\\sqrt{330}[\/latex] is the irrational part. Though we could write this as [latex]1\\sqrt{330}[\/latex], but the product of [latex]1[\/latex] and any other number is just the number.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm12704\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=12704&theme=lumen&iframe_resize_id=ohm12704&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":21,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Contemporary Mathematics\",\"author\":\"Donna Kirk\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/3-5-irrational-numbers\",\"project\":\"3.5 Irrational Numbers\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/1-introduction\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":53,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Contemporary Mathematics","author":"Donna Kirk","organization":"OpenStax","url":"https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/3-5-irrational-numbers","project":"3.5 Irrational Numbers","license":"cc-by","license_terms":"Access for free at https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/1-introduction"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8656"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":21,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8656\/revisions"}],"predecessor-version":[{"id":14161,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8656\/revisions\/14161"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/53"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8656\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8656"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8656"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8656"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8656"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}