// Numbas version: finer_feedback_settings
{"name": "1.2.4. Negative Indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "1.2.4. Negative Indices", "tags": [], "metadata": {"description": "<p>Part of HELM Book 1.2</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<h3>Negative Indices</h3>\n<p>Sometimes a number is raised to a negative power. This is interpreted as follows:</p>\n<div style=\"border-style: solid; border-color: black; border-radius: 25px; padding-left: 10px;\">\n<h4 style=\"text-align: center;\">Key Point 9</h4>\n<h3 style=\"text-align: center;\">Negative Powers</h3>\n<p style=\"text-align: center;\">\\( a^{-m}=\\frac{1}{a^m}\\qquad a^m = \\frac{1}{a^{-m}} \\)</p>\n</div>\n<p></p>\n<p>Thus a negative index can be used to indicate a reciprocal.</p>\n<p><strong>Example 21</strong></p>\n<p style=\"padding-left: 40px;\">Write each of the following expressions using a positive index and simplify if possible.</p>\n<p style=\"padding-left: 40px;\">(a) $2^{-3}$,$\\quad$ (b)$\\frac{1}{4^{-3}}$,$\\quad$ (c) $x^{-1}$, $\\quad$ (d) $x^{-2}$, $\\quad$ (e) $10^{-1}$</p>\n<p><strong>Solution</strong></p>\n<p style=\"padding-left: 40px;\">(a) $2^{-3}=\\frac{1}{2^3}=\\frac{1}{8}$,</p>\n<p style=\"padding-left: 40px;\">(b) $\\frac{1}{4^{-3}} = 4^3 = 64$</p>\n<p style=\"padding-left: 40px;\">(c) $x^{-1} = \\frac{1}{x^1} = \\frac{1}{x}$,</p>\n<p style=\"padding-left: 40px;\">(d) $x^{-2}=\\frac{1}{x^2}$,</p>\n<p style=\"padding-left: 40px;\">(e) $10^{-1}=\\frac{1}{10^1}=\\frac{1}{10}$ or $0.1$.</p>", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q1_expr": {"name": "q1_expr", "group": "question 1", "definition": "if(q1_1on,\n  expression(q1_letter + \"^-\" + q1_idx),\n  expression(\"1/\" + q1_letter + \"^-\" + q1_idx)\n)", "description": "", "templateType": "anything", "can_override": false}, "q1_idx": {"name": "q1_idx", "group": "question 1", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "alphabet": {"name": "alphabet", "group": "Ungrouped variables", "definition": "['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','v','w','x','y','z']", "description": "", "templateType": "anything", "can_override": false}, "q1_letter": {"name": "q1_letter", "group": "question 1", "definition": "random(random(alphabet),random(2..20))", "description": "", "templateType": "anything", "can_override": false}, "q1_1on": {"name": "q1_1on", "group": "question 1", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "q1_ans": {"name": "q1_ans", "group": "question 1", "definition": "if(q1_1on,\n  expression(\"1/\" + q1_letter + \"^\" + q1_idx),\n  expression(q1_letter + \"^\" + q1_idx)\n)", "description": "", "templateType": "anything", "can_override": false}, "q2_expr": {"name": "q2_expr", "group": "question 2", "definition": "expression( \n  \"(\" + q2_letter + \"^\" + q2_idx1 + \"*\" + q2_letter + \"^\" + q2_idx2 + \")\" +\n  \"/\" + q2_letter + \"^\" + q2_idx3 )", "description": "", "templateType": "anything", "can_override": false}, "q2_letter": {"name": "q2_letter", "group": "question 2", "definition": "random(random(alphabet),random(2..20))", "description": "", "templateType": "anything", "can_override": false}, "q2_idx1": {"name": "q2_idx1", "group": "question 2", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q2_idx2": {"name": "q2_idx2", "group": "question 2", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q2_idx3": {"name": "q2_idx3", "group": "question 2", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q2_idx": {"name": "q2_idx", "group": "question 2", "definition": "q2_idx1+q2_idx2-q2_idx3", "description": "", "templateType": "anything", "can_override": false}, "q2_ans": {"name": "q2_ans", "group": "question 2", "definition": "simplify(if(q2_idx > 0,\n  expression(q2_letter+\"^\"+q2_idx),\n  expression(\"1/\"+q2_letter+\"^\"+q2_negidx)\n),[\"basic\",\"unitPower\",\"zeroPower\",\"simplifyFractions\"]) ", "description": "", "templateType": "anything", "can_override": false}, "q2_negidx": {"name": "q2_negidx", "group": "question 2", "definition": "-q2_idx", "description": "", "templateType": "anything", "can_override": false}, "q3_expr": {"name": "q3_expr", "group": "Question 3", "definition": "expression(\"10^-\"+q3_idx)", "description": "", "templateType": "anything", "can_override": false}, "q3_ans": {"name": "q3_ans", "group": "Question 3", "definition": "rational(1/(10^(q3_idx)))", "description": "", "templateType": "anything", "can_override": false}, "q3_idx": {"name": "q3_idx", "group": "Question 3", "definition": "random(0..6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["alphabet"], "variable_groups": [{"name": "question 1", "variables": ["q1_idx", "q1_letter", "q1_1on", "q1_ans", "q1_expr"]}, {"name": "question 2", "variables": ["q2_letter", "q2_idx1", "q2_idx2", "q2_idx3", "q2_idx", "q2_negidx", "q2_expr", "q2_ans"]}, {"name": "Question 3", 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