{"id":"cb_question_bank:287ea7de-0444-4945-a3d0-8a9f23d040dd","source_id":"cb_question_bank","source_item_id":"287ea7de-0444-4945-a3d0-8a9f23d040dd","external_id":"aed0b60c-1ac2-47c0-8e11-047b2c3d1b5e","ibn":null,"question_id":"137cc6fd","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"H","score_band":7,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"RootIndex 5 StartRoot 70 n EndRoot left parenthesis RootIndex 6 StartRoot 70 n EndRoot right parenthesis squared For what value of x is the given expression equivalent to left parenthesis 70 n right parenthesis Superscript 30 x , where n greater than 1 ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"RootIndex 5 StartRoot 70 n EndRoot left parenthesis RootIndex 6 StartRoot 70 n EndRoot right parenthesis squared\"><mroot><mrow><mrow><mn>70</mn></mrow><mi>n</mi></mrow><mrow><mn>5</mn></mrow></mroot><msup><mfenced><mroot><mrow><mrow><mn>70</mn></mrow><mi>n</mi></mrow><mrow><mn>6</mn></mrow></mroot></mfenced><mn>2</mn></msup></math></p>\n<p style=\"text-align: left;\">For what value of <math alttext=\"x\"><mi>x</mi>\n</math> is the given expression equivalent to <math alttext=\"left parenthesis 70 n right parenthesis Superscript 30 x\"><msup><mfenced><mrow><mrow><mn>70</mn></mrow><mi>n</mi></mrow></mfenced><mrow><mrow><mn>30</mn></mrow><mi>x</mi></mrow></msup></math>, where <math alttext=\"n greater than 1\"><mi>n</mi><mo>&gt;</mo><mn>1</mn></math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"StartFraction 4 Over 225 EndFraction\"><mfrac>\n\t<mn>4</mn>\n\t<mn>225</mn>\n</mfrac>\n</math>. An expression of the form <math alttext=\"RootIndex k StartRoot a EndRoot\"><mroot><mi>a</mi><mi>k</mi></mroot></math>, where <math alttext=\"k\"><mi>k</mi>\n</math> is an integer greater than <math alttext=\"1\"><mn>1</mn>\n</math> and <math alttext=\"a greater than or equals 0\"><mi>a</mi><mo>≥</mo><mn>0</mn></math>, is equivalent to <math alttext=\"a Superscript StartFraction 1 Over k EndFraction\"><msup><mi>a</mi><mfrac><mn>1</mn><mi>k</mi></mfrac></msup></math>. Therefore, the given expression, where <math alttext=\"n greater than 1\"><mi>n</mi><mo>&gt;</mo><mn>1</mn></math>,&nbsp;is equivalent to <math alttext=\"left parenthesis 70 n right parenthesis Superscript one fifth Baseline left parenthesis left parenthesis 70 n right parenthesis Superscript one sixth Baseline right parenthesis squared\"><msup><mfenced><mrow><mn>70</mn><mi>n</mi></mrow></mfenced><mfrac><mn>1</mn><mn>5</mn></mfrac></msup><msup><mfenced><msup><mfenced><mrow><mn>70</mn><mi>n</mi></mrow></mfenced><mfrac><mn>1</mn><mn>6</mn></mfrac></msup></mfenced><mn>2</mn></msup></math>. Applying properties of exponents, this expression can be rewritten as&nbsp;<math alttext=\"left parenthesis 70 n right parenthesis Superscript one fifth Baseline left parenthesis 70 n right parenthesis Superscript one sixth dot 2\"><msup><mfenced><mrow><mn>70</mn><mi>n</mi></mrow></mfenced><mfrac><mn>1</mn><mn>5</mn></mfrac></msup><msup><mfenced><mrow><mn>70</mn><mi>n</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>·</mo><mn>2</mn></mrow></msup></math>, or&nbsp;<math alttext=\"left parenthesis 70 n right parenthesis Superscript one fifth Baseline left parenthesis 70 n right parenthesis Superscript one third\"><msup><mfenced><mrow><mn>70</mn><mi>n</mi></mrow></mfenced><mfrac><mn>1</mn><mn>5</mn></mfrac></msup><msup><mfenced><mrow><mn>70</mn><mi>n</mi></mrow></mfenced><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></math>, which can be rewritten as&nbsp;<math alttext=\"left parenthesis 70 n right parenthesis Superscript one fifth plus one third\"><msup><mfenced><mrow><mn>70</mn><mi>n</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>5</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup></math>, or&nbsp;<math alttext=\"left parenthesis 70 n right parenthesis Superscript eight fifteenths\"><msup><mfenced><mrow><mn>70</mn><mi>n</mi></mrow></mfenced><mfrac><mn>8</mn><mn>15</mn></mfrac></msup></math>. It's given that the expression&nbsp;<math alttext=\"RootIndex 5 StartRoot 70 n EndRoot left parenthesis RootIndex 6 StartRoot 70 n EndRoot right parenthesis squared\"><mroot><mrow><mn>70</mn><mi>n</mi></mrow><mn>5</mn></mroot><msup><mfenced><mroot><mrow><mn>70</mn><mi>n</mi></mrow><mn>6</mn></mroot></mfenced><mn>2</mn></msup></math> is equivalent to <math alttext=\"left parenthesis 70 n right parenthesis Superscript 30 x\"><msup><mfenced><mrow><mn>70</mn><mi>n</mi></mrow></mfenced><mrow><mn>30</mn><mi>x</mi></mrow></msup></math>, where <math alttext=\"n greater than 1\"><mi>n</mi><mo>&gt;</mo><mn>1</mn></math>. It follows that&nbsp;<math alttext=\"left parenthesis 70 n right parenthesis Superscript eight fifteenths\"><msup><mfenced><mrow><mn>70</mn><mi>n</mi></mrow></mfenced><mfrac><mn>8</mn><mn>15</mn></mfrac></msup></math> is equivalent to&nbsp;<math alttext=\"left parenthesis 70 n right parenthesis Superscript 30 x\"><msup><mfenced><mrow><mn>70</mn><mi>n</mi></mrow></mfenced><mrow><mn>30</mn><mi>x</mi></mrow></msup></math>. Therefore,&nbsp;<math alttext=\"eight fifteenths equals 30 x\"><mfrac><mn>8</mn><mn>15</mn></mfrac><mo>=</mo><mn>30</mn><mi>x</mi></math>. Dividing both sides of this equation by <math alttext=\"30\"><mn>30</mn>\n</math> yields&nbsp;<math alttext=\"StartFraction 8 Over 450 EndFraction equals x\"><mfrac><mn>8</mn><mn>450</mn></mfrac><mo>=</mo><mi>x</mi></math>, or <math alttext=\"StartFraction 4 Over 225 EndFraction equals x\"><mfrac><mn>4</mn><mn>225</mn></mfrac><mo>=</mo><mi>x</mi></math>. Thus, the value of <math alttext=\"x\"><mi>x</mi>\n</math> for which the given expression is equivalent to <math alttext=\"left parenthesis 70 n right parenthesis Superscript 30 x\"><msup><mfenced><mrow><mn>70</mn><mi>n</mi></mrow></mfenced><mrow><mn>30</mn><mi>x</mi></mrow></msup></math>, where&nbsp;<math alttext=\"n greater than 1\"><mi>n</mi><mo>&gt;</mo><mn>1</mn></math>, is <math alttext=\"StartFraction 4 Over 225 EndFraction\"><mfrac>\n\t<mn>4</mn>\n\t<mn>225</mn>\n</mfrac>\n</math>. Note that 4/225, .0177, .0178, 0.017, and 0.018 are examples of ways to enter a correct answer.</p>","correct_answer":[".0177",".0178","4/225"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.821000+00:00","source_updated_at":"2023-08-02T20:25:59.821000+00:00","options":[],"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}}