{"id":92,"date":"2017-08-23T07:47:31","date_gmt":"2017-08-23T07:47:31","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=92"},"modified":"2017-09-18T16:07:02","modified_gmt":"2017-09-18T16:07:02","slug":"laws-of-fractional-exponents","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/laws-of-fractional-exponents\/","title":{"rendered":"Laws of Fractional 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\/mathematics\/laws-of-integer-exponents\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-ii\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/exponential-functions\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Laws of Fractional Exponents<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study fractional exponents and the basic laws that govern how we add, subtract, multiply,\u00a0and divide them. Also, we will discuss the radical as a popular type of fractional exponent.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>The <em><strong>zero property\u00a0<\/strong><\/em>of exponents states that any number or variable raised to the\u00a0power of zero is equal to one.<\/li>\n<li>The <strong><em>power of\u00a0products property<\/em><\/strong> requires that the power is\u00a0distributed to each term in the product.<\/li>\n<li>The <em><strong>power of a power<\/strong><\/em> property is\u00a0used when an exponent is raised to a power. It states that when\u00a0an exponent is raised to a power, we must multiply the exponent\u00a0and the power.<\/li>\n<\/ul>\n<section>\n<h3>How do we perform operations on fractional exponents?<\/h3>\n<h4><strong>Adding and Subtracting<\/strong><\/h4>\n<p>We can only add or subtract like terms. The coefficients in\u00a0front of the variable will combine, while the exponent will remain\u00a0the same.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the correct simplification of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image003.gif\" width=\"45\" height=\"25\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image006.gif\" width=\"22\" height=\"25\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image009.gif\" width=\"22\" height=\"25\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image012.gif\" width=\"22\" height=\"25\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image015.gif\" width=\"13\" height=\"25\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is A. First, we must determine if the terms\u00a0being added are like terms. Because each term has the same\u00a0variable, with the same exponent, we know that these are like\u00a0terms. We then must add the coefficients of each term, resulting\u00a0in\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image018.gif\" width=\"22\" height=\"25\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the correct simplification of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image021.gif\" width=\"132\" height=\"25\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image024.gif\" width=\"61\" height=\"25\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image027.gif\" width=\"71\" height=\"25\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image030.gif\" width=\"71\" height=\"25\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image033.gif\" width=\"61\" height=\"25\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D. Again, we have to determine if we\u00a0have like terms, and if so combine them. Any remaining terms stay\u00a0separate. Since all terms\u00a0have <em>x\u00a0<\/em>as the variable, we look for terms with equivalent exponents.\u00a0There are two pairs of like terms, so we combine the two\u00a0pairs and get the following result.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image036.gif\" width=\"93\" height=\"25\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image039.gif\" width=\"122\" height=\"25\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>We have combined all the like terms. Therefore, the correct\u00a0choice is D.<\/p>\n<\/div>\n<\/section>\n<h4>Multiplying and Dividing<\/h4>\n<p>When multiplying two or more terms with fractional exponents,\u00a0the exponents are added together and the coefficients are\u00a0multiplied. When dividing two or more terms, the difference of the\u00a0exponents is found, and the coefficients are divided.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the correct simplification of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image042.gif\" width=\"106\" height=\"28\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/>?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image045.gif\" width=\"32\" height=\"25\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image048.gif\" width=\"31\" height=\"25\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image051.gif\" width=\"30\" height=\"25\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image054.gif\" width=\"24\" height=\"25\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is C. First, we multiply the coefficients\u00a0together: (2)(4)(1) = 8. We then add the exponents.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image057.gif\" width=\"165\" height=\"40\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Our final answer is \u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image060.gif\" width=\"30\" height=\"25\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Which choice shows the correct simplification of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image063.gif\" width=\"32\" height=\"64\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0?<\/p>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image066.gif\" width=\"47\" height=\"28\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image069.gif\" width=\"54\" height=\"28\" name=\"graphics25\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image072.gif\" width=\"37\" height=\"28\" name=\"graphics26\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image075.gif\" width=\"43\" height=\"28\" name=\"graphics27\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<table border=\"2\" width=\"75%\" cellpadding=\"10\" align=\"center\">\n<tbody>\n<tr>\n<td>\n<p class=\"notebox_text\" align=\"CENTER\"><strong>Important Tidbit<\/strong><\/p>\n<p class=\"notebox_text\" align=\"CENTER\">Remember that for negative exponents, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image078.gif\" width=\"45\" height=\"34\" name=\"graphics28\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is B.\u00a0We divide each term with the same variable, and subtract the\u00a0corresponding exponents. The variable <em>z\u00a0<\/em>is then multiplied.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image081.gif\" width=\"92\" height=\"49\" name=\"graphics29\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image084.gif\" width=\"102\" height=\"52\" name=\"graphics30\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The terms combine, and result in\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image087.gif\" width=\"24\" height=\"64\" name=\"graphics31\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This answer can be rewritten with negative exponents as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p2_clip_image090.gif\" width=\"54\" height=\"28\" name=\"graphics32\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h3>The Radical<\/h3>\n<p>Earlier, we discussed radical symbols and how they can be\u00a0converted into a variable raised to the one-half power. In other\u00a0words,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p3_clip_image003.gif\" width=\"50\" height=\"26\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>The variable <em>x<\/em>,\u00a0when it stands alone or when it is beneath the radical, is raised\u00a0to the first power, which accounts for the 1 in the numerator of\u00a0the fractional exponent. The 2 in the denominator is derived from\u00a0the fact that this is a <em>square\u00a0root<\/em>.<\/p>\n<p>Cube roots can also be converted\u00a0to fractional exponents. In this case,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s8_p3_clip_image006.gif\" width=\"50\" height=\"26\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>We can continue this pattern to\u00a0convert any variation of a radical raised to a power to a\u00a0fractional exponent.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/s8_p3_html_m7414dc37.gif\" width=\"101\" height=\"75\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Let&#8217;s try an example.\u00a0Multiply the radicals\u00a0<img decoding=\"async\" class=\"no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/s8_p3_html_m3194687d.gif\" name=\"graphics6\" \/>.<\/p>\n<p><strong>Step 1:<\/strong> Convert each radical to its equivalent\u00a0fractional exponent.<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/s8_p3_html_63c8bbbd.gif\" width=\"64\" height=\"64\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/center><strong>Step 2:<\/strong> Multiple the fractional exponents.<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/s8_p3_html_29a39c65.gif\" width=\"81\" height=\"33\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/center><strong>Step 3:<\/strong> Convert back into radical form.<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/s8_p3_html_m66bc89b.gif\" width=\"73\" height=\"35\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/center><\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/laws-of-integer-exponents\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-ii\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/exponential-functions\">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 Laws of Fractional Exponents Objective In this lesson, you will study fractional exponents and the basic laws that govern how we add, subtract, multiply,\u00a0and divide them. Also, we will discuss the radical as a popular type of fractional exponent. Previously Covered: The zero property\u00a0of exponents states that any number [&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-92","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/92","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=92"}],"version-history":[{"count":8,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/92\/revisions"}],"predecessor-version":[{"id":746,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/92\/revisions\/746"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=92"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}