// Numbas version: exam_results_page_options
{"name": "1.2.1. Index notation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "1.2.1. Index notation", "tags": [], "metadata": {"description": "<p>Part of HELM Book 1.2</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<h3>1. Index notation</h3>\n<p>The number $4 \\times 4 \\times 4$ is written, for short, as $4^3$ and read &lsquo;$4$ raised to the power $3$&rsquo; or &lsquo;$4$ cubed&rsquo;. Note that the number of times &lsquo;$4$&rsquo; occurs in the product is written as a superscript. In this context we call the superscript $3$ an <strong>index</strong> or <strong>power</strong>. Similarly we could write</p>\n<p style=\"text-align: center;\">$5 \\times 5 = 5^2,\\quad$ read &lsquo;$5$ to the power $2$&rsquo; or &lsquo;$5$ squared&rsquo;</p>\n<p>and</p>\n<p style=\"text-align: left; padding-left: 40px;\">$7 \\times 7 \\times 7 \\times 7 \\times 7 = 7^5\\;, \\quad a \\times a \\times a = a^3\\;,\\quad m\\times m\\times m \\times m = m^4$</p>\n<p style=\"text-align: left;\">More generally, in the expression $x^y$ , $x$ is called the <strong>base</strong> and $y$ is called the index or power. The plural of index is <strong>indices</strong>. The process of raising to a power is also known as <strong>exponentiation </strong>because yet another name for a power is an <strong>exponent</strong>. When dealing with numbers your calculator is able to evaluate expressions involving powers, probably using the $x^y$ button.</p>\n<h4 style=\"text-align: left;\"><strong>Example 12</strong></h4>\n<p style=\"text-align: left; padding-left: 40px;\">Use a calculator to evaluate $3^{12}$</p>\n<h4 style=\"text-align: left;\"><strong>Solution</strong></h4>\n<p style=\"text-align: left; padding-left: 40px;\">Using the $x^y$ button on the calculator check that you obtain $3^{12}=531441$.</p>\n<h4 style=\"text-align: left;\"><strong>Example 13</strong></h4>\n<p style=\"text-align: left; padding-left: 40px;\">Identify the index and base in the following expressions. (a) $8^{11}$,$\\quad$ (b) $(-2)^5$, $\\quad$ (c) $p^{-q}$</p>\n<h4 style=\"text-align: left;\"><strong>Solution</strong></h4>\n<p style=\"text-align: left; padding-left: 40px;\">(a) In the expression $8^{11}$, $8$ is the base and $11$ is the index.</p>\n<p style=\"text-align: left; padding-left: 40px;\">(b) In the expression $(-2)^5$, $-2$ is the base and $5$ is the index.</p>\n<p style=\"text-align: left; padding-left: 40px;\">(c) In the expression $p^{-q}$, $p$ is the base and $-q$ is the index. The interpretation of a negative index will be given in sub-section 4 of this book.</p>\n<p style=\"text-align: left;\">Recall from Section 1.1 that when several operations are involved we can make use of the BODMAS rule for deciding the order in which operations must be carried out. The BODMAS rule makes no mention of exponentiation. Exponentiation should be carried out immediately after any brackets have been dealt with and before multiplication and division. Consider the following examples.</p>\n<h4 style=\"text-align: left;\"><strong>Example 14</strong></h4>\n<p style=\"text-align: left; padding-left: 40px;\">Evaluate $7\\times 3^2$.</p>\n<h4 style=\"text-align: left;\"><strong>Solution</strong></h4>\n<p style=\"text-align: left; padding-left: 40px;\">There are two operations involved here, exponentiation and multiplication. The exponentiation should be carried out before the multiplication. So $7\\times 3^2 = 7 \\times 9 = 63$.</p>\n<h4 style=\"text-align: left;\"><strong>Example 15&nbsp;</strong></h4>\n<p style=\"text-align: left; padding-left: 40px;\">Write out fully (a) $3m^4\\;,\\quad$ (b) $(3m)^4$.</p>\n<h4 style=\"text-align: left;\"><strong>Solution</strong></h4>\n<p style=\"text-align: left; padding-left: 40px;\">(a) In the expression $3m^4$ the exponentiation is carried out before the multiplication by $3$. So</p>\n<p style=\"text-align: left; padding-left: 80px;\">\\(3m^4=3\\times(m\\times m\\times m\\times m) = 3\\times m\\times m \\times m \\times m\\)</p>\n<p style=\"text-align: left; padding-left: 40px;\">(b) Here the bracketed expression is raised to the power $4$ and so should be multiplied by itself four times:</p>\n<p style=\"text-align: left; padding-left: 80px;\">\\((3m)^4 = (3m)\\times (3m)\\times (3m)\\times (3m)\\)</p>\n<p style=\"padding-left: 40px;\">Because of the associativity of multiplication we can write this as</p>\n<p style=\"padding-left: 40px;\">\\( 3\\times 3\\times 3\\times 3 \\times m \\times m \\times m \\times m = 81m^4. \\)</p>\n<p style=\"text-align: left; padding-left: 40px;\">Note the important distinction between $(3m)^4$ and $3m^4$.</p>", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"base1": {"name": "base1", "group": "Q1", "definition": "random(2,3)", 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