{"id":18011,"date":"2025-06-12T20:11:53","date_gmt":"2025-06-12T20:11:53","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=18011"},"modified":"2025-06-12T20:12:01","modified_gmt":"2025-06-12T20:12:01","slug":"which-monomial-is-a-perfect-cube-2","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/which-monomial-is-a-perfect-cube-2\/","title":{"rendered":"Which monomial is a perfect cube"},"content":{"rendered":"\n<p>Which monomial is a perfect cube? 16&#215;6 27&#215;8 32&#215;12 64&#215;6<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n<p>Let&#8217;s analyze each monomial to determine which one is a <strong>perfect cube<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Given monomials:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>16x616x^6<\/li>\n\n\n\n<li>27x827x^8<\/li>\n\n\n\n<li>32x1232x^{12}<\/li>\n\n\n\n<li>64x664x^6<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Understand what makes a monomial a perfect cube<\/h3>\n\n\n\n<p>A <strong>perfect cube<\/strong> means the entire monomial can be expressed as (a\u22c5xb)3(a \\cdot x^b)^3, where both the numerical coefficient aa and the variable term xbx^b are perfect cubes individually.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The coefficient must be a perfect cube (e.g., 1,8,27,64,125,\u20261, 8, 27, 64, 125, \\ldots)<\/li>\n\n\n\n<li>The exponent on the variable must be divisible by 3 because (xk)3=x3k(x^k)^3 = x^{3k}<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Check each monomial<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">1. 16x616x^6<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Coefficient: 16\n<ul class=\"wp-block-list\">\n<li>16 is not a perfect cube because 23=82^3 = 8, 33=273^3 = 27, so 16 is not a perfect cube.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Exponent of xx: 6\n<ul class=\"wp-block-list\">\n<li>6 is divisible by 3, so x6=(x2)3x^6 = (x^2)^3 is a perfect cube term.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>Since the coefficient is not a perfect cube, <strong>16x^6<\/strong> is <strong>not<\/strong> a perfect cube.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">2. 27x827x^8<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Coefficient: 27\n<ul class=\"wp-block-list\">\n<li>27=3327 = 3^3, so this is a perfect cube.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Exponent of xx: 8\n<ul class=\"wp-block-list\">\n<li>8 is not divisible by 3, so x8x^8 is not a perfect cube term.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>Therefore, <strong>27x^8<\/strong> is <strong>not<\/strong> a perfect cube.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">3. 32x1232x^{12}<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Coefficient: 32\n<ul class=\"wp-block-list\">\n<li>32=2532 = 2^5, which is not a perfect cube (since 23=82^3=8, 26=642^6=64)<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Exponent of xx: 12\n<ul class=\"wp-block-list\">\n<li>12 is divisible by 3, so x12=(x4)3x^{12} = (x^4)^3 is a perfect cube term.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>Since the coefficient is not a perfect cube, <strong>32x^{12}<\/strong> is <strong>not<\/strong> a perfect cube.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">4. 64x664x^6<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Coefficient: 64\n<ul class=\"wp-block-list\">\n<li>64=4364 = 4^3 or 262^6, so 64 is a perfect cube.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Exponent of xx: 6\n<ul class=\"wp-block-list\">\n<li>6 is divisible by 3, so x6=(x2)3x^6 = (x^2)^3 is a perfect cube.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>Since both the coefficient and variable term are perfect cubes, <strong>64x^6<\/strong> <strong>is<\/strong> a perfect cube.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Final answer:<\/strong><\/h3>\n\n\n\n<p>64&#215;6\\boxed{64x^6}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation (about 300 words):<\/h3>\n\n\n\n<p>A monomial is a perfect cube if it can be written as some expression raised to the third power. For example, (axb)3=a3x3b(a x^b)^3 = a^3 x^{3b}. This means two conditions must be met:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>The coefficient (numerical part) must itself be a perfect cube.<\/li>\n\n\n\n<li>The exponent on the variable must be divisible by 3 because when you cube xbx^b, the exponent becomes 3b3b.<\/li>\n<\/ol>\n\n\n\n<p>Let&#8217;s apply this to the four given monomials:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>16x\u2076<\/strong>: The coefficient 16 is not a perfect cube since 23=82^3 = 8 and 33=273^3 = 27, but 16 falls in between. Although the variable part x6x^6 can be expressed as (x2)3(x^2)^3, the coefficient prevents this monomial from being a perfect cube.<\/li>\n\n\n\n<li><strong>27x\u2078<\/strong>: Here, 27 is a perfect cube (333^3), but the exponent on xx is 8, which is not divisible by 3, so it cannot be expressed as (xk)3(x^k)^3.<\/li>\n\n\n\n<li><strong>32x^{12}<\/strong>: The variable part is perfect cube because 12 is divisible by 3, but 32 is 252^5, not a perfect cube. So this monomial is not a perfect cube.<\/li>\n\n\n\n<li><strong>64x\u2076<\/strong>: 64 is a perfect cube because 43=644^3 = 64. Also, 6 is divisible by 3, so x6=(x2)3x^6 = (x^2)^3. This monomial can be written as (4&#215;2)3(4x^2)^3, confirming it is a perfect cube.<\/li>\n<\/ul>\n\n\n\n<p>Therefore, the only monomial that is a perfect cube from the list is <strong>64x^6<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Which monomial is a perfect cube? 16&#215;6 27&#215;8 32&#215;12 64&#215;6 The correct answer and explanation is: Let&#8217;s analyze each monomial to determine which one is a perfect cube. Given monomials: Step 1: Understand what makes a monomial a perfect cube A perfect cube means the entire monomial can be expressed as (a\u22c5xb)3(a \\cdot x^b)^3, where [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-18011","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/18011","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=18011"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/18011\/revisions"}],"predecessor-version":[{"id":18012,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/18011\/revisions\/18012"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=18011"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=18011"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=18011"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}