{"id":12766,"date":"2014-01-31T21:58:23","date_gmt":"2014-01-31T21:58:23","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=12766"},"modified":"2026-03-25T12:46:38","modified_gmt":"2026-03-25T17:46:38","slug":"roots","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/roots\/","title":{"rendered":"Roots"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_3PwVOSg2dWQ\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_3PwVOSg2dWQ\" data-source-videoID=\"3PwVOSg2dWQ\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Roots Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Roots\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_3PwVOSg2dWQ:hover {cursor:pointer;} img#videoThumbnailImage_3PwVOSg2dWQ {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/1194-roots-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_3PwVOSg2dWQ\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_3PwVOSg2dWQ\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Roots\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_3PwVOSg2dWQ\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_3PwVOSg2dWQ\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_3PwVOSg2dWQ\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction CjQ_Function() {\n  var x = document.getElementById(\"CjQ\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"CjQ_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"CjQ\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Understanding_Terminology\" class=\"smooth-scroll\">Understanding Terminology<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Notation\" class=\"smooth-scroll\">Notation<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Practice_Problems\" class=\"smooth-scroll\">Practice Problems<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Frequently_Asked_Questions\" class=\"smooth-scroll\">Frequently Asked Questions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Root_Practice_Questions\" class=\"smooth-scroll\">Root Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"FAQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"FAQs\">FAQs<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video on roots! Today, we will be working towards an understanding of the terminology, notation, and interpretation of algebraic roots. We will also be making connections to other concepts that you will need in higher-level math. Let\u2019s get started!<\/p>\n<h2><span id=\"Understanding_Terminology\" class=\"m-toc-anchor\"><\/span>Understanding Terminology<\/h2>\n<p>\nUnderstanding the terminology and notation of math is half the battle if you\u2019re struggling to grasp certain concepts. This is true of roots, where the terminology that is used determines the \u201ctype\u201d of root that is evaluated. <\/p>\n<h3><span id=\"Finding_the_Square_Root\" class=\"m-toc-anchor\"><\/span>Finding the Square Root<\/h3>\n<p>\nTo find the <strong>square root<\/strong> of a number, simply ask yourself, \u201cwhat value, when <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/multiplying-and-dividing-fractions\/\">multiplied<\/a> by itself, results in that number?\u201d  <\/p>\n<p>For example, let\u2019s say you were asked to find the square root of 4. Ask yourself, \u201cwhat value, when multiplied by itself, results in 4?\u201d The answer is 2, because 2 times 2 equals 4.<\/p>\n<p>Let\u2019s try another one. What is the square root of 121? Ask yourself, \u201cwhat value, when multiplied by itself, results in 121?\u201d The answer is 11, because 11 times 11 equals 121.<\/p>\n<h3><span id=\"Finding_the_Cube_Root\" class=\"m-toc-anchor\"><\/span>Finding the Cube Root<\/h3>\n<p>\nTo find the <strong>cube root<\/strong> of a number, ask yourself, \u201cwhat value, when multiplied by itself <strong>three<\/strong> times, results in that number?\u201d<\/p>\n<p>For example, the cube root of 8 would be 2, because 2 times itself three times equals 8. The cube root of 64 is 4, because 4 times itself three times equals 8. Four times four is 16, 16 times 4 is 64.<\/p>\n<div class=\"examplesentence\">\\(2 \\cdot 2 \\cdot 2=8\\)<br \/>\n\\(4 \\cdot 4=16\\)<br \/>\n\\(16 \\cdot4=64\\)<\/div>\n<p>\n&nbsp;<br \/>\nFourth roots, fifth roots, sixth roots, and so on, can be found similarly.  <\/p>\n<h3><span id=\"Perfect_Squares\" class=\"m-toc-anchor\"><\/span>Perfect Squares<\/h3>\n<p>\nThere is an important relationship revealed in these practice problems. We just showed that 2 is the square root of 4. That means that the number 4 is a perfect square. Knowing the <strong>perfect squares<\/strong> from 1 through 144 is helpful in order to simplify radicals in the future. The table here shows these perfect squares as related to their square roots.<\/p>\n<table class=\"ATable\" style=\"margin: auto;\">\n<thead>\n<tr>\n<th colspan=\"3\"><strong>Perfect Squares<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\\(1^2=1\\)<\/td>\n<td>\\(5^2=25\\)<\/td>\n<td>\\(9^2=81\\)<\/td>\n<\/tr>\n<tr>\n<td>\\(2^2=4\\)<\/td>\n<td>\\(6^2=36\\)<\/td>\n<td>\\(10^2=100\\)<\/td>\n<\/tr>\n<tr>\n<td>\\(3^2=9\\)<\/td>\n<td>\\(7^2=49\\)<\/td>\n<td>\\(11^2=121\\)<\/td>\n<\/tr>\n<tr>\n<td>\\(4^2=16\\)<\/td>\n<td>\\(8^2=64\\)<\/td>\n<td>\\(12^2=144\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Perfect_Cubes\" class=\"m-toc-anchor\"><\/span>Perfect Cubes<\/h3>\n<p>\n<strong>Perfect cubes<\/strong> can also be quickly determined by multiplying any integer by itself three times. In one of the previous examples, we showed that 4 is the cube root of 64. This means that 64 is a perfect cube.<\/p>\n<table class=\"ATable\" style=\"margin: auto;\">\n<thead>\n<tr>\n<th colspan=\"2\"><strong>Perfect Cubes<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width:50%\">\\(1^3=1\\)<\/td>\n<td style=\"width:50%\">\\(6^3=216\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width:50%\">\\(2^3=8\\)<\/td>\n<td style=\"width:50%\">\\(7^3=343\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width:50%\">\\(3^3=27\\)<\/td>\n<td style=\"width:50%\">\\(8^3=512\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width:50%\">\\(4^3=64\\)<\/td>\n<td style=\"width:50%\">\\(9^3=729\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width:50%\">\\(5^3=125\\)<\/td>\n<td style=\"width:50%\">\\(10^3=1,000\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nAs revealed in the table above, perfect cubes increase rapidly! <\/p>\n<h2><span id=\"Notation\" class=\"m-toc-anchor\"><\/span>Notation<\/h2>\n<h3><span id=\"Radical\" class=\"m-toc-anchor\"><\/span>Radical<\/h3>\n<p>\nTo generalize the rule of finding roots, let\u2019s introduce the notation of <strong>radicals<\/strong>. Let\u2019s break down this notation into \u201cparts\u201d by looking at the cube root of 27, which looks like this: \\(\\sqrt[3]{27}\\).<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/07\/roots-and-such.png\" alt=\"\" width=\"279.3\" height=\"149.3\" class=\"aligncenter size-full wp-image-86599\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/07\/roots-and-such.png 419w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/07\/roots-and-such-300x160.png 300w\" sizes=\"(max-width: 419px) 100vw, 419px\" \/><\/p>\n<p>The radical may resemble a division symbol, but it has a very different meaning. <\/p>\n<h3><span id=\"Radicand\" class=\"m-toc-anchor\"><\/span>Radicand<\/h3>\n<p>\nWhat is under the radical symbol is called the <strong>radicand<\/strong>, and this can be a number or an algebraic expression. We are going to stick to numbers in this video.<\/p>\n<h3><span id=\"Index\" class=\"m-toc-anchor\"><\/span>Index<\/h3>\n<p>\nThe <strong>index<\/strong> is the most important feature. That small number placed in the \u201ccheckmark\u201d of the radical symbol indicates the root. In this example, because the index is 3 they are asking for the cube root of 27. With a bit of thought, we can determine that 3 times 3 times 3 equals 27, so the cube root of 27 is 3, which means that 27 is a perfect cube.<\/p>\n<p>It is important to note that the square root symbol does not show an index of 2. So just remember that when there is NO index indicated, the radical represents a square root by default.<\/p>\n<h3><span id=\"Exponent_Notation\" class=\"m-toc-anchor\"><\/span>Exponent Notation<\/h3>\n<p>\nFor our final point in this video, let\u2019s make an important link from radical notation to <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/addition-and-subtraction-with-exponents\/\">exponen<\/a> notation.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/07\/equation-labeled.png\" alt=\"\" width=\"363.24\" height=\"153.72\" class=\"aligncenter size-full wp-image-86602\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/07\/equation-labeled.png 1009w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/07\/equation-labeled-300x127.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/07\/equation-labeled-768x325.png 768w\" sizes=\"(max-width: 1009px) 100vw, 1009px\" \/><\/p>\n<p>A radical is the same as raising a base to a fractional exponent, where the index of the radical becomes the denominator of the fractional exponent and the exponent of the radicand is the numerator of the fractional exponent. Here is a generalized example:<\/p>\n<p>Let\u2019s say that the radicand of a radical is \\(x^a\\), and the index of the radical is \\(b\\). This is equivalent to raising a base, \\(x\\), to the fractional exponent of <span style=\"font-size:120%\">\\(\\frac{a}{b}\\)<\/span>.<\/p>\n<h2><span id=\"Practice_Problems\" class=\"m-toc-anchor\"><\/span>Practice Problems<\/h2>\n<p>\nPracticing a few examples will help to make some sense of all of this notation and terminology.  <\/p>\n<h3><span id=\"Problem_1\" class=\"m-toc-anchor\"><\/span>Problem #1<\/h3>\n<p>\nFirst, let\u2019s practice expressing a radical as a fractional exponent: \\(\\sqrt[3]{x^2}\\)<\/p>\n<p>Remember, the index of the radical becomes the denominator of the fractional exponent, which in this case is 3. The exponent of the radicand becomes the numerator of the fractional exponent, which is 2 in this case. So, our fractional exponent is \\(x^{\\frac{2}{3}}\\):<\/p>\n<div class=\"examplesentence\">\\(\\sqrt[3]{x^2}=x^{\\frac{2}{3}}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Problem_2\" class=\"m-toc-anchor\"><\/span>Problem #2<\/h3>\n<p>\nLet\u2019s try another one: \\(\\sqrt[4]{3}\\)<\/p>\n<p>So we know that our index, 4, becomes the denominator, but the radicand doesn\u2019t have an exponent. That means that 1 will become our numerator, which gives us \\(3^{\\frac{1}{4}}\\):<\/p>\n<div class=\"examplesentence\">\\(\\sqrt[4]{3}=3^{\\frac{1}{4}}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow let\u2019s flip things around and convert a fractional exponent to a radical. Our fractional exponent is \\(125^{\\frac{1}{3}}\\).<\/p>\n<p>If we reverse what we were doing before, the denominator of the fractional exponent becomes the index of our radical. Since the numerator here is 1, we don\u2019t have an exponent for the radicand, so we end up with \\(\\sqrt[3]{125}\\):<\/p>\n<div class=\"examplesentence\">\\(125^{\\frac{1}{3}}=\\sqrt[3]{125}\\)<\/div>\n<p>\n&nbsp;<br \/>\nOnce the conversion to a radical is made, the problem becomes more familiar and the root is easier to evaluate: The cube root of 125 is 5.<\/p>\n<p>I hope this review was helpful! Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"FAQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Frequently_Asked_Questions\" class=\"m-toc-anchor\"><\/span>Frequently Asked Questions<\/h2>\n<div class=\"faq-list\">\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is a root in math?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>The <strong>root<\/strong> of a number in math is a number that when multiplied by itself produces the original number. For example, the square root of 49 is 7 because \\(7\\times7=49\\). In this case, because 7 is multiplied by itself twice to produce49, we call 7 the <strong>square root<\/strong> of 49.<\/p>\n<p>The cube root of 27 is 3 because \\(3\\times3\\times3=27\\). Since 3 is multiplied three times to produce 27 we call this a cube root, so 3 is the <strong>cube root<\/strong> of 27.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">How do you find the roots in math?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>To find the root of a number in math, we start by finding the factors of that number. For example, the factors of 64 are \\(2\\times2\\times2\\times2\\times2\\times2\\). If we look a bit closer, we see the factors can also be written as \\(8\\times8\\):<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Roots-FAQ-1.png\" alt=\"2x2x2x2x2x2 grouped into 2 even groups\" width=\"345.9\" height=\"86.7\" class=\"aligncenter size-full wp-image-106317\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Roots-FAQ-1.png 1153w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Roots-FAQ-1-300x75.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Roots-FAQ-1-1024x257.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Roots-FAQ-1-768x192.png 768w\" sizes=\"(max-width: 1153px) 100vw, 1153px\" \/><br \/>So we know that the square root of 64 is 8 because \\(8\\times8=64\\). Since 8 is being multiplied by itself <em>twice<\/em>, we call this the square root of 64.<\/p>\n<p>We can also bunch the factors into three groups:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Roots-FAQ-2.png\" alt=\"2x2x2x2x2x2 grouped into 3 even groups\" width=\"345\" height=\"86.7\" class=\"aligncenter size-full wp-image-106320\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Roots-FAQ-2.png 1161w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Roots-FAQ-2-300x75.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Roots-FAQ-2-1024x255.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/12\/Roots-FAQ-2-768x191.png 768w\" sizes=\"(max-width: 1161px) 100vw, 1161px\" \/><br \/>This means \\(4\\times4\\times4\\) also equals 64. Since 4 is multiplied by itself three times to get 64, we know that 4 is the cube root of 64.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What does a \\(\\sqrt{}\\) mean in math?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>This is the symbol that represents square root. The square root of a number is the number when multiplied by itself produces the original number. For example, the square root of 16, or \\(\\sqrt{16}\\), is 4, because \\(4\\times4=16\\).<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is the easiest way to find cube roots?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>The <strong>cube root<\/strong> of a number is a number that is multiplied by itself three times to give the original number.<\/p>\n<p>The easiest way to find the cube root of a number is to start by finding the factors and see if there are three numbers in the factor that are the same. <\/p>\n<div class=\"lightbulb-example-2\"><span class=\"lightbulb-icon\">\ud83d\udca1<\/span><span class=\"faq-example-question\">Example: Find the cube root of 125.<\/span><\/p>\n<hr style=\"padding: 0; margin-top: -0.2em; margin-bottom: 1.2em\">First, find the factors, which are \\(5\\times5\\times5\\).<\/p>\n<p style=\"margin-bottom: 0em\">Since 5 is being multiplied by itself three times to produce 125, we can say that 5 is the cube root of 125.<\/p>\n<\/div>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is a radical in math?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>A radical in math is the symbol \\(\\sqrt{ }\\), which is used to represent a root. If there is no index (number in the \u201carm\u201d of the radical), then it is assumed to be a square root.<\/p>\n<p>To represent the expression \u201csquare root of 36\u201d, we place the 36 under the radical: \\(\\sqrt{36}\\).<\/p>\n<p>A square root is a number that when multiplied by itself will produce the original number under the radical. Therefore, the square root of 36 is equal to 6 because \\(6\\times6=36\\). This can also be written as \\(\\sqrt{36}=6\\).<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">How do you solve a radical?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>To solve a radical, which represents a square root, we start by finding the factors of the number that is under the radical. <\/p>\n<div class=\"lightbulb-example-2\"><span class=\"lightbulb-icon\">\ud83d\udca1<\/span><span class=\"faq-example-question\">Example:  Solve \\(\\sqrt{49}\\).<\/span><\/p>\n<hr style=\"padding: 0; margin-top: -0.2em; margin-bottom: 1.2em\"> First, we find the factors of 49, which are \\(7\\times7\\). Since 7 is being multiplied by itself twice, we can conclude that 7 is the square root of 49. <\/p>\n<p style=\"margin-bottom: 0em\">Therefore, \\(\\sqrt{49}=7\\).<\/p>\n<\/div>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is a radical simplified?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>To simplify a radical, you must find the square root of the number until nothing under the radical has any roots. <\/p>\n<div class=\"lightbulb-example-2\"><span class=\"lightbulb-icon\">\ud83d\udca1<\/span><span class=\"faq-example-question\">Example: Simplify the radical \\(\\sqrt{18}\\).<\/span><\/p>\n<hr style=\"padding: 0; margin-top: -0.2em; margin-bottom: 1.2em\"> Start by finding the factors of 18, which are \\(3\\times3\\times2\\).<\/p>\n<p>Since 3 is multiplied twice by itself, we can pull that root out and the 3 would go in front of the radical, while the 2 remains under the radical. <\/p>\n<p style=\"text-align: center\">\\(\\sqrt{18}=3\\sqrt{2}\\)<\/p>\n<p style=\"margin-bottom: 0em\">This is the most simplified form of the expression because you cannot simplify \\(\\sqrt{2}\\).<\/p>\n<\/div>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is an example of a radical number?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>A <strong>radical<\/strong> is a symbol that represents square root. The number under the radical is called a <strong>radicand<\/strong>.<\/p>\n<p>For example, the expression \u201cthe square root of 81\u201d is represented in math by the radical symbol with 81 under the radical. The \\(\\sqrt{81}=9\\) because \\(9\\times9=81\\).<\/p>\n<p>The symbol is the radical, 81 is the radicand, and 9 is the root.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is a radicand of a square root?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>The <strong>radicand<\/strong> is the number under the radical that we are trying to find the root for. For example, the \u201csquare root of 100\u201d can be written as \\(\\sqrt{100}\\). The number under the symbol, which is called the radical, is called the radicand. In this case, 100 is the radicand.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is the index and radicand?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>The <strong>index<\/strong> is the root that we are trying to find, and the <strong>radicand<\/strong> is the number under the radical symbol.<\/p>\n<p>For example, \\(\\sqrt{25}\\) is the square root of 25. There is an imaginary 2 that we don&#8217;t write, which tells us that we should be taking the square root of the number. In this case, 2 is the index and 25 is the radicand.<\/p>\n<p>The expression \\(\\sqrt[\\displaystyle 3]{64}\\), is the cube root of 64. The 3 is the index, the 64 is the radicand, and the square root symbol is called the radical.<\/p>\n<p>The index tells us which root of the radicand we are supposed to find. For the square root of 25, we are finding a number that is multiplied by itself twice to get 25. For the cube root of 64, we are looking for a number that is multiplied by itself three times to get 64.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is the difference between a radical and a radicand?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>The <strong>radical<\/strong> is the symbol that represents square root. The <strong>radicand<\/strong> is the number that is under the radical that we are trying to find the root of. For example, in the expression \\(\\sqrt{50}\\), the symbol is the radical and 50, which is under the radical symbol, is called the radicand.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What is an example of a radicand?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>A <strong>radicand<\/strong> is the number under the radical. In the expression \\(\\sqrt{36}\\), the 36 is the radicand because it is under the radical, which is the square root symbol.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Root_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Root Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhat is \\(\\sqrt[3]{x^4}\\) expressed as a fractional exponent?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">\\(x^\\frac{3}{4}\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\(x^\\frac{2}{3}\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\(x^\\frac{3}{2}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-4\">\\(x^\\frac{4}{3}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The correct answer is \\(x^\\frac{4}{3}\\). When turning a root into a fractional exponent, the number in the hook of the radical symbol, 3 in this case, becomes the denominator of the fractional exponent. The power, 4, of the base, x, becomes the numerator. The base stays the same, so \\(\\sqrt[3]{x^4}=x^\\frac{4}{3}\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhat is \\(x^\\frac{7}{2}\\) expressed as a radical?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\(\\sqrt[3]{x^4}\\)<\/div><div class=\"PQ\"  id=\"PQ-2-2\">\\(\\sqrt[4]{x^3}\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-3\">\\(\\sqrt{x^7}\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(\\sqrt[7]{x^2}\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The correct answer is \\(\\sqrt{x^7}\\). When turning a fractional exponent into a radical, the numerator of the fraction becomes the exponent the base is raised to inside a radical symbol with the denominator of the fraction as the index. So \\(x^\\frac{7}{2}=\\sqrt[2]{x^7}\\), which can also be written as \\(\\sqrt{x^7}\\) because no index is always assumed to be 2.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\n\\(\\sqrt{256}=\\)<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-3-1\">16<\/div><div class=\"PQ\"  id=\"PQ-3-2\">15<\/div><div class=\"PQ\"  id=\"PQ-3-3\">14<\/div><div class=\"PQ\"  id=\"PQ-3-4\">13<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The correct answer is 16. To find the square root of 256, ask what number times itself gives you 256. \\(16\u00d716=256\\), so \\(\\sqrt{256}=16\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\n\\(\\sqrt[3]{64}=\\)<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">9<\/div><div class=\"PQ\"  id=\"PQ-4-2\">6<\/div><div class=\"PQ\"  id=\"PQ-4-3\">3<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-4\">4<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The correct answer is 4. To find the cube root of 64, ask what number times itself three times gives you 64. \\(4\u00d74\u00d74=64\\), so \\(\\sqrt[3]{64}=4\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhat is the name of the small number placed in the \u201ccheckmark\u201d of the radical symbol?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">Radicand<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-2\">Index<\/div><div class=\"PQ\"  id=\"PQ-5-3\">Radical<\/div><div class=\"PQ\"  id=\"PQ-5-4\">Cube<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The correct answer is index. The index is the small number place in the \u201ccheckmark\u201d of the radical symbol and indicates the root.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/basic-arithmetic\/\">Return to Basic Arithmetic Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Basic Arithmetic Videos<\/p>\n","protected":false},"author":1,"featured_media":100309,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-12766","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-math-advertising-group","7":"page_category-pre-algebra-operations-videos","8":"page_category-radicals","9":"page_category-video-pages-for-study-course-sidebar-ad","10":"page_type-video","11":"content_type-practice-questions","12":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/12766","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=12766"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/12766\/revisions"}],"predecessor-version":[{"id":238636,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/12766\/revisions\/238636"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100309"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=12766"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}