{"id":"cb_question_bank:d246eaab-7366-4aa3-b329-734096f5c394","source_id":"cb_question_bank","source_item_id":"d246eaab-7366-4aa3-b329-734096f5c394","external_id":null,"ibn":"08224-DC","question_id":"e117d3b8","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"If a and c are positive numbers, which of the following is equivalent to the square root of, open parenthesis, a plus c, close parenthesis, cubed, end root, times, the square root of a plus c, end root ?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">If <span class=\"italic\">a</span> and <span class=\"italic\">c</span> are positive numbers, which of the following is equivalent to <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f11a2734d08f15f1c740c7343bedf5cae5b2e05f2a599d99400e7d56aa65677c\" alt=\"the square root of, open parenthesis, a plus c, close parenthesis, cubed, end root, times, the square root of a plus c, end root\"></span></span>?</p>\n","rationale_html":"<p>Choice C is correct. Using the property that&nbsp;<span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ba9fd6f09c7cda41527e5bb6c8ed85f77aa33886b02aee0951e98376f1abf246\" alt=\"the square root of x, end root, times the square root of y, end root, equals the square root of x y, end root\"></span> for positive numbers <span class=\"italic\">x</span> and <span class=\"italic\">y</span>, with <span class=\"italic\">x</span> = (<span class=\"italic\">a </span>+<span class=\"italic\"> c</span>)<sup>3</sup> and <span class=\"italic\">y </span>= <span class=\"italic\">a</span> + <span class=\"italic\">c</span>, it follows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2e45c0f8ab641393ce4f98a1b7aeb979b8cbd0feb8825db52da8214db63f73e9\" alt=\"the square root of, open parenthesis, a, plus c, close parenthesis, cubed, end root, times, the square root of a, plus c, end root, equals the square root of, open parenthesis a, plus c, end parenthesis to the power 4, end root\"></span>. By rewriting (<span class=\"italic\">a</span> + <span class=\"italic\">c</span>)<sup>4</sup> as ((<span class=\"italic\">a</span> + <span class=\"italic\">c</span>)<sup>2</sup>)<sup>2</sup>, it is possible to simplify the square root expression as follows: <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/97fe18e9e292f195075276745e4a281c53b0421e7fadf301670ec348073cf5e3\" alt=\"the square root of, open outer parenthesis, open inner parenthesis, a, plus c, close inner parenthesis, squared, close outer parenthesis, squared, end root, equals, open parenthesis, a, plus c, close parenthesis, squared, which equals a, squared, plus 2 a, c, plus c squared\"></span>.</p><p>\n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        \n        Choice A is incorrect and may be the result of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1273adca08791a87676d7ac2620f890b6c6c56de7637bd52570b1e0bee29db81\" alt=\"the square root of, open parenthesis, a, plus c, close parenthesis, cubed, end root, divided by the square root of,  a, plus c, end root\"></span>. Choice B is incorrect and may be the result of incorrectly rewriting (<span class=\"italic\">a </span>+ <span class=\"italic\">c</span>)<sup>2</sup> as <span class=\"italic\">a</span><sup>2</sup> + <span class=\"italic\">c</span><sup>2</sup>. Choice D is incorrect and may be the result of incorrectly applying properties of exponents.</p><p></p>\n","correct_answer":["C"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.635000+00:00","source_updated_at":"2023-08-02T20:25:59.635000+00:00","options":[{"label":"A","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/154fab496336ba4065b27485cd84bd149fe6a335fedc5b792d113c7244f2ee87\" alt=\"a plus c\"></span></span>\n","ord":0},{"label":"B","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f9244d175edf1e285a508f7733ab33908cf77cfac1b150c088a0f266ce22bccb\" alt=\"a squared plus c squared\"></span></span>\n","ord":1},{"label":"C","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cf2e0dcbc39afde983c2dec4ba91c4f8a574deb515375dc9853a884dcbaac473\" alt=\"a squared plus 2 a c plus c squared\"></span></span>\n","ord":2},{"label":"D","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0aaa13d58231b460201e13e1935ba8c7dd06d8418b19b9dc9d1ca0cdf91de224\" alt=\"a squared times c squared\"></span></span>\n","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}