{"id":44627,"date":"2025-06-30T17:57:10","date_gmt":"2025-06-30T17:57:10","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=44627"},"modified":"2025-06-30T17:57:11","modified_gmt":"2025-06-30T17:57:11","slug":"simplify-each-cube-root-expression","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/simplify-each-cube-root-expression\/","title":{"rendered":"Simplify each cube root expression"},"content":{"rendered":"\n
Simplify each cube root expression. Describe the simplified form of the expression as rational or irrational. In your final answer, include all of your work. 3 square root of 81 3 square root of -64<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
Problem 1: Simplify the cube root of 81.<\/h3>\n\n\n\n
The expression we are asked to simplify is 813\\sqrt[3]{81}381\u200b.<\/p>\n\n\n\n
Step 1: Express 81 in terms of its prime factors.<\/h4>\n\n\n\n
We start by factoring 81: 81=3481 = 3^481=34<\/p>\n\n\n\n
Step 2: Apply the cube root.<\/h4>\n\n\n\n
The cube root of a number xxx, denoted x3\\sqrt[3]{x}3x\u200b, is the number that, when multiplied by itself three times, equals xxx. Now, we apply the cube root to 81=3481 = 3^481=34: 813=343\\sqrt[3]{81} = \\sqrt[3]{3^4}381\u200b=334\u200b<\/p>\n\n\n\n
We can rewrite this as: 343=343\\sqrt[3]{3^4} = 3^{\\frac{4}{3}}334\u200b=334\u200b<\/p>\n\n\n\n
Step 3: Simplify the exponent.<\/h4>\n\n\n\n
To simplify 3433^{\\frac{4}{3}}334\u200b, we separate the exponent into two parts: 343=31+13=31\u00d73133^{\\frac{4}{3}} = 3^{1 + \\frac{1}{3}} = 3^1 \\times 3^{\\frac{1}{3}}334\u200b=31+31\u200b=31\u00d7331\u200b 31=3and33=3133^1 = 3 \\quad \\text{and} \\quad \\sqrt[3]{3} = 3^{\\frac{1}{3}}31=3and33\u200b=331\u200b<\/p>\n\n\n\n
Therefore, the simplified expression is: 813=3\u00d733\\sqrt[3]{81} = 3 \\times \\sqrt[3]{3}381\u200b=3\u00d733\u200b<\/p>\n\n\n\n
Conclusion for Problem 1:<\/h4>\n\n\n\n
The cube root of 81 simplifies to 3\u00d7333 \\times \\sqrt[3]{3}3\u00d733\u200b. Since 33\\sqrt[3]{3}33\u200b is irrational, the entire expression is irrational. Therefore, the final answer is 3\u00d7333 \\times \\sqrt[3]{3}3\u00d733\u200b.<\/p>\n\n\n\n
\n\n\n\n
Problem 2: Simplify the cube root of -64.<\/h3>\n\n\n\n
We are given the expression \u2212643\\sqrt[3]{-64}3\u221264\u200b.<\/p>\n\n\n\n
Step 1: Express -64 as a product of prime factors.<\/h4>\n\n\n\n
The prime factorization of 64 is: 64=2664 = 2^664=26<\/p>\n\n\n\n
Since we have a negative sign, we can write: \u221264=\u22121\u00d764=\u22121\u00d726-64 = -1 \\times 64 = -1 \\times 2^6\u221264=\u22121\u00d764=\u22121\u00d726<\/p>\n\n\n\n
Step 2: Apply the cube root.<\/h4>\n\n\n\n
Now, we find the cube root of \u221264-64\u221264: \u2212643=\u22121\u00d7263\\sqrt[3]{-64} = \\sqrt[3]{-1 \\times 2^6}3\u221264\u200b=3\u22121\u00d726\u200b<\/p>\n\n\n\n
We can separate this into two parts: \u221213\u00d7263\\sqrt[3]{-1} \\times \\sqrt[3]{2^6}3\u22121\u200b\u00d7326\u200b<\/p>\n\n\n\n
Step 3: Simplify each part.<\/h4>\n\n\n\n
The cube root of -1 is \u22121-1\u22121, and the cube root of 262^626 is: 263=22=4\\sqrt[3]{2^6} = 2^2 = 4326\u200b=22=4<\/p>\n\n\n\n
Thus, the simplified expression is: \u2212643=\u22121\u00d74=\u22124\\sqrt[3]{-64} = -1 \\times 4 = -43\u221264\u200b=\u22121\u00d74=\u22124<\/p>\n\n\n\n
Conclusion for Problem 2:<\/h4>\n\n\n\n
The cube root of -64 simplifies to \u22124-4\u22124, which is a rational number.<\/p>\n\n\n\n
Final Answer:<\/h3>\n\n\n\n\n
813=3\u00d733\\sqrt[3]{81} = 3 \\times \\sqrt[3]{3}381\u200b=3\u00d733\u200b, which is irrational<\/strong>.<\/li>\n\n\n\n
\u2212643=\u22124\\sqrt[3]{-64} = -43\u221264\u200b=\u22124, which is rational<\/strong>.<\/li>\n<\/ol>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"
Simplify each cube root expression. Describe the simplified form of the expression as rational or irrational. In your final answer, include all of your work. 3 square root of 81 3 square root of -64 The Correct Answer and Explanation is: Problem 1: Simplify the cube root of 81. The expression we are asked to […]<\/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-44627","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/44627","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=44627"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/44627\/revisions"}],"predecessor-version":[{"id":44631,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/44627\/revisions\/44631"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=44627"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=44627"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=44627"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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