{"id":210,"date":"2017-08-22T16:30:52","date_gmt":"2017-08-22T16:30:52","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/elementary-education\/?page_id=210"},"modified":"2017-09-22T15:56:48","modified_gmt":"2017-09-22T15:56:48","slug":"working-with-powers-and-exponents","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/elementary-education\/working-with-powers-and-exponents\/","title":{"rendered":"Working with Powers and Exponents"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/fractions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/number-sense\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/proportional-reasoning\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Working with Powers and Exponents<\/h1>\n<h4>Objective<\/h4>\n<p>This next section will help you brush up on exponents and fractional exponents. This will also be a refresher on roots and scientific notation.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>In the previous section, we reviewed and practiced the basic operations with mixed numbers and other fractions.<\/li>\n<\/ul>\n<section>\n<h3>Exponents Rule!<\/h3>\n<p>No, wait, that should be exponent rules. This section is filled with all the rules you need to know for dealing with numerical exponents.<\/p>\n<p>First, let&#8217;s review the terminology:<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0060M.gif\" alt=\"Base 2, exponent 5\" \/><\/center><\/p>\n<p style=\"text-align: center;\">Two is the base. Five is the exponent. The whole thing (2<sup>5<\/sup>) is called a power.<\/p>\n<p>Now we&#8217;ll start with the basic rules.<\/p>\n<table>\n<thead>\n<tr>\n<th colspan=\"3\">Basic Rules of Exponents<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Rule<\/strong><\/td>\n<td><strong>Definition<\/strong><\/td>\n<td><strong>Example<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Raising to the First Power<\/td>\n<td>For any non-zero number a, a<sup>1<\/sup> = a.<\/td>\n<td>6<sup>1<\/sup> = 6<\/td>\n<\/tr>\n<tr>\n<td>Raising to the Zero Power<\/td>\n<td>For any non-zero number a, a<sup>0<\/sup> = 1, by definition.<\/td>\n<td>6<sup>0<\/sup> = 1<\/td>\n<\/tr>\n<tr>\n<td>Negative Exponents<\/td>\n<td>For any non-zero number a and integer n, a<sup>-n<\/sup> = 1\/a<sup>n<\/sup>.<\/td>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0061M.gif\" alt=\"Ex 1\" \/><br \/>\n<img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0062M.gif\" alt=\"Ex 2\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Here is one for you to practice on your own.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Write the following expression with positive exponents: <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0066M.gif\" alt=\"Question\" \/><\/p>\n<ol>\n<li><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0064M.gif\" alt=\"Answer 1\" \/><\/li>\n<li><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0065M.gif\" alt=\"Answer 2\" \/><\/li>\n<li><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0063S.gif\" alt=\"Answer 3\" \/><\/li>\n<li><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0067M.gif\" alt=\"Answer 4\" \/><\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">The correct answer is C. The a<sup>2<\/sup> term stays in the denominator because its exponent is positive. The b<sup>-4<\/sup> term must move into the denominator to make its exponent positive. The c<sup>3<\/sup> term must move into the denominator to make its exponent positive, which gives you a final answer of <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0063S.gif\" alt=\"Answer\" \/>.<\/p>\n<\/section>\n<p>Here are the more <strong>complex rules<\/strong>.<\/p>\n<table>\n<thead>\n<tr>\n<th colspan=\"3\">Advanced Rules<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Rule<\/strong><\/td>\n<td><strong>Definition<\/strong><\/td>\n<td><strong>Example<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Multiplying Powers with the Same Base<\/td>\n<td>For any non-zero number a and integers m and n, <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0081b.gif\" alt=\"Same base multiplication\" \/>.<\/td>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0082a.gif\" alt=\"Example\" \/><\/p>\n<p>(Add the exponents when multiplying numbers with the same base.)<\/td>\n<\/tr>\n<tr>\n<td>Raising a Power to a Power<\/td>\n<td>For any non-zero number a and integers m and n, (a<sup>m<\/sup>)<sup>n<\/sup> = a<sup>mn<\/sup>.<\/td>\n<td>(x<sup>5<\/sup>)<sup>2<\/sup> = x<sup>10<\/sup><\/p>\n<p>(Multiply the exponents when a power is raised to a power.)<\/td>\n<\/tr>\n<tr>\n<td>Raising a Product to a Power<\/td>\n<td>or any non-zero numbers a and b, and integer n, (ab)<sup>n<\/sup> = a<sup>n<\/sup> x b<sup>n<\/sup>.<\/td>\n<td><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0084a.gif\" alt=\"Example\" \/><\/p>\n<p>(Each part of the product is separately raised to the power.)<\/td>\n<\/tr>\n<tr>\n<td>Raising a Quotient to a Power<\/td>\n<td>For any non-zero numbers a and b, and integer n, (a\/b)<sup>n<\/sup> = a<sup>n<\/sup>\/b<sup>n<\/sup><\/td>\n<td>(x\/5)<sup>3<\/sup> = x<sup>3<\/sup>\/5<sup>3<\/sup> = x<sup>3<\/sup>\/125<\/p>\n<p>(Each part of the quotient is separately raised to the power.)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When simplifying expressions with exponents, numbers raised to a power should be multiplied out.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Simplify the expression (3d)<sup>4<\/sup>(d<sup>2<\/sup>)<sup>3<\/sup><\/p>\n<ol>\n<li>3d<sup>10<\/sup><\/li>\n<li>12d<sup>9<\/sup><\/li>\n<li>27d<sup>9<\/sup><\/li>\n<li>81d<sup>10<\/sup><\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">The correct answer is D. (3<i>d<\/i>)<sup>4<\/sup> is equal to 3<sup>4<\/sup><i>d<\/i><sup>4<\/sup>, or 81<i>d<\/i><sup>4<\/sup>. (<i>d<\/i><sup>2<\/sup>)<sup> 3<\/sup> is equal to <i>d<\/i><sup>6<\/sup> because you multiply the powers when you raise a power to a power. Then add the powers when you multiply 81<i>d<\/i><sup>4<\/sup> \u2022 <i>d<\/i><sup>6<\/sup>, to get 81<i>d<\/i><sup>10<\/sup><\/p>\n<\/section>\n<h3>Roots<\/h3>\n<p>The opposite operation of raising a number to a power is to take its root. Square roots are the root with which we are most familiar. Taking the square root is essentially removing the square from the number.<\/p>\n<h4>Example<\/h4>\n<p>6<sup>2<\/sup> = 36<\/p>\n<p>\u221a36 = 6<\/p>\n<p>(Technically, \u221a36 can also be (-6), since (-6)<sup>2<\/sup> also equals 36.)<\/p>\n<p>To find roots other than the square root, follow the same process. Without a calculator, you can really only find perfect roots. You do this by working backwards and using mental math. To find the \u221a36, you have to think, some number times itself equals 36. Algebraically, it would look like:<\/p>\n<p>x \u2022 x = 36<\/p>\n<p>For a cube root, it is the same, only there are three Xs instead of two.<\/p>\n<h4>Example<\/h4>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0072M.gif\" alt=\"Example 1\" \/><\/center>For higher numbered roots, the number of Xs is the same as the number of the root.<\/p>\n<h4>Example<\/h4>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0073M.gif\" alt=\"Example 2\" \/><\/center>Generally, you will only be asked to do fairly simple problems on this topic; it&#8217;s better suited for a calculator than for mental math.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Solve <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0074M.gif\" alt=\"3rd root of 125\" \/><\/p>\n<ol>\n<li>5<\/li>\n<li>15<\/li>\n<li>25<\/li>\n<li>41<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">The correct choice is A. 5 \u2022 5 \u2022 5 is 125. On problems like this, you can always work backwards from the answer choices.<\/p>\n<\/section>\n<h3>Fractional Powers<\/h3>\n<p>Finding fractional powers are like finding roots. Fractional powers require a bit of logical thinking, but, for our purposes, we can examine some problems that only require some brainpower.<\/p>\n<p>Briefly, here is the theory behind fractional powers. Let&#8217;s look at it with the help of an example.<\/p>\n<p class=\"center\">16<sup>1\/2<\/sup> = x<\/p>\n<p class=\"center\">16<sup>1\/2<\/sup> \u2022 16<sup>1\/2<\/sup> = 16<sup>1\/2+1\/2<\/sup> = 16<sup>1<\/sup><\/p>\n<p>Utilize the rule of multiplying powers with the same base.<\/p>\n<p>Now go back and replace 16<sup>1\/2<\/sup> with x.<\/p>\n<p class=\"center\">x \u2022 x = 16<sup>1<\/sup><\/p>\n<p>So x = 4, which means that 16<sup>1\/2<\/sup> = 4. That is the same as the square root of 16.<\/p>\n<p>How about 49<sup>1\/2<\/sup>?<\/p>\n<p>You should have said 7.<\/p>\n<p>Other fractional powers with a numerator of 1 work the same way, so a power of 1\/3 is just like finding the cube root of the given number.<\/p>\n<h4>Example<\/h4>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0075L.gif\" alt=\"Example\" \/><\/center>If the numerator is a number other than 1, it tells you how many of the root number you need. If we raise 27 to the <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0076MP.gif\" alt=\"2\/3\" \/> power, we still find the cube root, which is 3, but we need it twice. Multiply 3 x 3 and you get 9.<\/p>\n<h4>Example<\/h4>\n<p>32<sup>3\/5<\/sup> = ?<\/p>\n<p><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0077MP.gif\" alt=\"32 to the 1\/5 power\" \/> = 2<\/p>\n<p>2 \u2022 2 \u2022 2 = 2<sup>3<\/sup> = 8<\/p>\n<p>So, 32<sup>3\/5<\/sup> = 8<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Solve <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0079MP.gif\" alt=\"64 to the 1\/3 power\" \/><\/p>\n<ol>\n<li>2<\/li>\n<li>4<\/li>\n<li>8<\/li>\n<li>12<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">The correct answer is B. Finding a number to the <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/eq0080MP.gif\" alt=\"1\/3\" \/> power is the same as finding the cube root of the number, and 4 is the cube root of 64.<\/p>\n<\/section>\n<h3>When Am I Ever Going to Use This?<\/h3>\n<p>Exponents can seem a bit esoteric; they aren&#8217;t exactly the sort of math you need to use to balance your checkbook. But scientific notation is a fairly common use of powers and exponents.<\/p>\n<p>Scientific notation is especially useful for expressing very large numbers (think astronomy) and very small numbers (think microbiology).<\/p>\n<p>Numbers written in scientific notation are composed of two parts. The first part is a number between 1 and 10 (it doesn&#8217;t have to be a whole number, many are decimal numbers). There can only be one digit to the left of the decimal, so the largest number is really 9.999&#8230;. not 10. The second part is 10 raised to a power, which can be negative or positive. The two parts are multiplied together, so that they look like these examples:<\/p>\n<p class=\"center\">3.24 x 10<sup>5<\/sup><\/p>\n<p class=\"center\">9.7021 x 10<sup>-3<\/sup><\/p>\n<h3>So how does this work?<\/h3>\n<h4>Convert from Scientific Notation to Standard Form<\/h4>\n<p>To write a number given in scientific notation as a number in standard form, you move the decimal point. The basic numbers of the notation do not change; the decimal moves, and you may need some placeholder zeroes.<\/p>\n<p>The number of places the decimal moves is equal to the exponent.<\/p>\n<p>For a positive exponent, move the decimal to the right\u2014you are making a <em>big<\/em> number.<\/p>\n<p>For a negative exponent, move the decimal to the left\u2014you are making a <em>small<\/em> number (But remember: You are not making a negative number.) Multiplying by 10<sup>\u22123<\/sup> is the same as dividing by 10<sup>3<\/sup>.<\/p>\n<h4>Examples<\/h4>\n<p>3.24 x 10<sup>5<\/sup> = 324,000 (move the decimal 5 places to the right, filling in the 3 extra places with zeroes)<\/p>\n<p>9.7021 x 10<sup>\u22123<\/sup> = 0.0097021 (move the decimal 3 places to the left, filling in the 2 extra places with zeroes)<\/p>\n<p>Now it is your turn to practice.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Write 7.059 x 10<sup>8<\/sup> in standard form.<\/p>\n<ol>\n<li>7.05900000<\/li>\n<li>705,000,900<\/li>\n<li>705,900,000<\/li>\n<li>705,900,000,000<\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">The correct answer is C. Move the decimal 8 places to the right, which means you will need 5 placeholder zeroes. Put the commas in to make the number easier to read. Note that choice A has the right number of zeroes, but the decimal point is in the wrong place.<\/p>\n<\/section>\n<h4>Convert from Standard Form to Scientific Notation<\/h4>\n<p>To write a number given in standard form as a number in scientific notation, there are two steps.<\/p>\n<p>First, write the decimal number. Remember, you can only have one place to the left of the decimal. You can have as many places to the right as necessary. Drop any placeholder zeroes.<\/p>\n<h4>Examples<\/h4>\n<p>517,000 becomes 5.17; 0.0004906 becomes 4.906<\/p>\n<p>Second, raise the 10 to the correct power. To determine the power, count how many places you needed to move the decimal to make your decimal number. The exponent will be positive if the standard form was a big number (greater than 1). The exponent is negative if the standard form of the number is between 0 and 1.<\/p>\n<h4>Examples<\/h4>\n<p>517,000 = 5.17 x 10<sup>5<\/sup> (the decimal has to move 5 places)<\/p>\n<p>0.0004906 = 4.906 x 10<sup>\u00ad4<\/sup> (the decimal has to move 4 places)<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>Write 0.00259 in scientific notation.<\/p>\n<ol>\n<li>25.9 x 10<sup>-4<\/sup><\/li>\n<li>2.59 x 10<sup>-3<\/sup><\/li>\n<li>0.259 x 10<sup>2<\/sup><\/li>\n<li>-2.59 x 10<sup>3<\/sup><\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">The correct answer is B. It has the decimal number properly formatted, and the number of the exponent is correctly counted. Choice D is similar, but the original number was positive, so the negative sign belongs to the exponent, not the decimal number.<\/p>\n<\/section>\n<h3>Review<\/h3>\n<p>There are a number of rules for working with exponents and powers. Most importantly:<\/p>\n<ul>\n<li>To multiply powers with the same base, add their exponents.<\/li>\n<li>To divide powers with the same base, subtract their exponents.<\/li>\n<li>To raise a power to a power, multiply the exponents.<\/li>\n<li>Any number raised to the zero power is one.<\/li>\n<li>The opposite operation of raising a number to a power is taking its root.<\/li>\n<li>Finding fractional powers of a number is the same as finding the root of a number: raising a number to the 1\/2 power is the same as finding its square root.<\/li>\n<li>Numbers in scientific notation are composed of a decimal number between 1 and 10 and a power of 10.<\/li>\n<li>Numbers between 0 and 1 have negative exponents.<\/li>\n<li>Numbers greater than 1 have positive exponents.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/fractions\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/number-sense\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/proportional-reasoning\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Working with Powers and Exponents Objective This next section will help you brush up on exponents and fractional exponents. This will also be a refresher on roots and scientific notation. Previously Covered: In the previous section, we reviewed and practiced the basic operations with mixed numbers and other fractions. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-210","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/210","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/comments?post=210"}],"version-history":[{"count":19,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/210\/revisions"}],"predecessor-version":[{"id":1309,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/210\/revisions\/1309"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/media?parent=210"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}