{"id":737,"date":"2024-04-25T15:43:02","date_gmt":"2024-04-25T15:43:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=737"},"modified":"2025-08-13T14:55:13","modified_gmt":"2025-08-13T14:55:13","slug":"radicals-and-rational-exponents-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/radicals-and-rational-exponents-apply-it-1\/","title":{"raw":"Radicals and Rational Exponents: Apply It 1","rendered":"Radicals and Rational Exponents: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Calculate square roots and apply them in basic operations like addition, subtraction, and rationalizing the denominators.<\/li>\r\n \t<li>Use the product and quotient rules to simplify expressions that include square roots.<\/li>\r\n \t<li>Understand how to use rational exponents in expressions and recognize their connection to roots for simplifying calculations.<\/li>\r\n<\/ul>\r\n<\/section>Quilt blocks are fabric components sewn together to make a quilt.\r\n\r\n[caption id=\"attachment_4200\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-4200\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25002953\/real_quilt_block-300x200.jpg\" alt=\"\" width=\"500\" height=\"333\" \/> Colorful elephant tapestries on display. Photo by Ibrahim Rifath on Unsplash[\/caption]\r\n\r\nThe quilt block design of a square within a square is called an \u201ceconomy quilt block.\u201d A textile artist is making a quilt using economy quilt blocks of varying sizes, so he makes a sketch of the design with side length [latex]w[\/latex]\u00a0as shown below.\r\n\r\n[caption id=\"attachment_7474\" align=\"aligncenter\" width=\"895\"]<img class=\"wp-image-7474 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20153927\/1.3.A.Diagram1-2-1.png\" alt=\"The image shows a geometric diagram with two parts. On the left is a square with width &quot;w&quot; containing nested shapes: a purple square rotated 45 degrees (appearing as a diamond) inside a blue square, all set against a green background. Each shape is labeled with measurements showing &quot;w\/2&quot; for various dimensions. On the right side is a smaller right triangle in green with sides labeled &quot;w\/2&quot;, followed by a question mark\" width=\"895\" height=\"584\" \/> Geometric breakdown of a square with nested shapes and right triangle dimensions labeled.[\/caption]\r\n\r\nThe artist needs to calculate how many square inches of fabric he will need for each orange triangle. First, he will need to determine an expression for the hypotenuse.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]25367[\/ohm2_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]25368[\/ohm2_question]<\/section>\r\n\r\n[caption id=\"attachment_7476\" align=\"aligncenter\" width=\"568\"]<img class=\"wp-image-7476 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20154005\/1.3.A.Diagram3-2.png\" alt=\"The image shows a green isosceles triangle with a dashed vertical line down its middle, dividing it into two equal parts. The triangle's base is labeled &quot;base&quot; across the bottom, its height is labeled &quot;height&quot; along the middle dashed line, and both sides are labeled &quot;w\/2&quot;. A small curved arrow is drawn inside the triangle near the dashed line, indicating an angle.\" width=\"568\" height=\"340\" \/> Triangle with labeled base, height, and halves of the base marked as \ud835\udc64\u2044\u2082.[\/caption]\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]25369[\/ohm2_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]25370[\/ohm2_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]25371[\/ohm2_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]25372[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate square roots and apply them in basic operations like addition, subtraction, and rationalizing the denominators.<\/li>\n<li>Use the product and quotient rules to simplify expressions that include square roots.<\/li>\n<li>Understand how to use rational exponents in expressions and recognize their connection to roots for simplifying calculations.<\/li>\n<\/ul>\n<\/section>\n<p>Quilt blocks are fabric components sewn together to make a quilt.<\/p>\n<figure id=\"attachment_4200\" aria-describedby=\"caption-attachment-4200\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4200\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25002953\/real_quilt_block-300x200.jpg\" alt=\"\" width=\"500\" height=\"333\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25002953\/real_quilt_block-300x200.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25002953\/real_quilt_block-1024x682.jpg 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25002953\/real_quilt_block-768x512.jpg 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25002953\/real_quilt_block-1536x1023.jpg 1536w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25002953\/real_quilt_block-65x43.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25002953\/real_quilt_block-225x150.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25002953\/real_quilt_block-350x233.jpg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25002953\/real_quilt_block.jpg 1600w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-4200\" class=\"wp-caption-text\">Colorful elephant tapestries on display. Photo by Ibrahim Rifath on Unsplash<\/figcaption><\/figure>\n<p>The quilt block design of a square within a square is called an \u201ceconomy quilt block.\u201d A textile artist is making a quilt using economy quilt blocks of varying sizes, so he makes a sketch of the design with side length [latex]w[\/latex]\u00a0as shown below.<\/p>\n<figure id=\"attachment_7474\" aria-describedby=\"caption-attachment-7474\" style=\"width: 895px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-7474 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20153927\/1.3.A.Diagram1-2-1.png\" alt=\"The image shows a geometric diagram with two parts. On the left is a square with width &quot;w&quot; containing nested shapes: a purple square rotated 45 degrees (appearing as a diamond) inside a blue square, all set against a green background. Each shape is labeled with measurements showing &quot;w\/2&quot; for various dimensions. On the right side is a smaller right triangle in green with sides labeled &quot;w\/2&quot;, followed by a question mark\" width=\"895\" height=\"584\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20153927\/1.3.A.Diagram1-2-1.png 895w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20153927\/1.3.A.Diagram1-2-1-300x196.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20153927\/1.3.A.Diagram1-2-1-768x501.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20153927\/1.3.A.Diagram1-2-1-65x42.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20153927\/1.3.A.Diagram1-2-1-225x147.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20153927\/1.3.A.Diagram1-2-1-350x228.png 350w\" sizes=\"(max-width: 895px) 100vw, 895px\" \/><figcaption id=\"caption-attachment-7474\" class=\"wp-caption-text\">Geometric breakdown of a square with nested shapes and right triangle dimensions labeled.<\/figcaption><\/figure>\n<p>The artist needs to calculate how many square inches of fabric he will need for each orange triangle. First, he will need to determine an expression for the hypotenuse.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm25367\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=25367&theme=lumen&iframe_resize_id=ohm25367&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm25368\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=25368&theme=lumen&iframe_resize_id=ohm25368&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<figure id=\"attachment_7476\" aria-describedby=\"caption-attachment-7476\" style=\"width: 568px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-7476 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20154005\/1.3.A.Diagram3-2.png\" alt=\"The image shows a green isosceles triangle with a dashed vertical line down its middle, dividing it into two equal parts. The triangle's base is labeled &quot;base&quot; across the bottom, its height is labeled &quot;height&quot; along the middle dashed line, and both sides are labeled &quot;w\/2&quot;. A small curved arrow is drawn inside the triangle near the dashed line, indicating an angle.\" width=\"568\" height=\"340\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20154005\/1.3.A.Diagram3-2.png 568w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20154005\/1.3.A.Diagram3-2-300x180.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20154005\/1.3.A.Diagram3-2-65x39.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20154005\/1.3.A.Diagram3-2-225x135.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/20154005\/1.3.A.Diagram3-2-350x210.png 350w\" sizes=\"(max-width: 568px) 100vw, 568px\" \/><figcaption id=\"caption-attachment-7476\" class=\"wp-caption-text\">Triangle with labeled base, height, and halves of the base marked as \ud835\udc64\u2044\u2082.<\/figcaption><\/figure>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm25369\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=25369&theme=lumen&iframe_resize_id=ohm25369&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm25370\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=25370&theme=lumen&iframe_resize_id=ohm25370&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm25371\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=25371&theme=lumen&iframe_resize_id=ohm25371&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm25372\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=25372&theme=lumen&iframe_resize_id=ohm25372&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":27,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":32,"module-header":"apply_it","content_attributions":[],"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/737"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":18,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/737\/revisions"}],"predecessor-version":[{"id":7524,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/737\/revisions\/7524"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/32"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/737\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=737"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=737"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=737"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=737"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}