{"id":646,"date":"2019-04-29T16:20:42","date_gmt":"2019-04-29T20:20:42","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/?post_type=chapter&p=646"},"modified":"2019-12-05T11:28:21","modified_gmt":"2019-12-05T16:28:21","slug":"8-2-multiplication-and-division-of-rational-expressions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/chapter\/8-2-multiplication-and-division-of-rational-expressions\/","title":{"raw":"8.2 Multiplication and Division of Rational Expressions","rendered":"8.2 Multiplication and Division of Rational Expressions"},"content":{"raw":"[latexpage]\r\n\r\nMultiplying and dividing rational expressions is very similar to the process used to multiply and divide fractions.\r\n
\r\n

Example 8.2.1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nReduce and multiply \\(\\dfrac{15}{49}\\) and \\(\\dfrac{14}{45}\\).\r\n\r\n\\[\\dfrac{15}{49}\\cdot \\dfrac{14}{45}\\text{ reduces to }\\dfrac{1}{7}\\cdot \\dfrac{2}{3}, \\text { which equals }\\dfrac{2}{21}\\]\r\n\r\n(15 and 45 reduce to 1 and 3, and 14 and 49 reduce to 2 and 7)\r\n\r\n<\/div>\r\n<\/div>\r\n

This process of multiplication is identical to division, except the first step is to reciprocate any fraction that is being divided. <\/span><\/p>\r\n\r\n

\r\n

Example 8.2.2<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nReduce and divide \\(\\dfrac{25}{18}\\) by \\(\\dfrac{15}{6}\\).\r\n

\\(\\dfrac{25}{18} \\div \\dfrac{15}{6} \\text{ reciprocates to } \\dfrac{25}{18}\\cdot \\dfrac{6}{15}, \\text{ which reduces to }\\dfrac{5}{3}\\cdot \\dfrac{1}{3}, \\text{ which equals } \\dfrac{5}{9}\\)<\/p>\r\n(25 and 15 reduce to 5 and 3, and 6 and 18 reduce to 1 and 3)\r\n\r\n<\/div>\r\n<\/div>\r\n

When multiplying with rational expressions,\u00a0 follow the same process: first, divide out common factors, then multiply straight across. <\/span><\/p>\r\n\r\n

\r\n

Example 8.2.3<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nReduce and multiply \\(\\dfrac{25x^2}{9y^8}\\) and \\(\\dfrac{24y^4}{55x^7}\\).\r\n\r\n\\[\\dfrac{25x^2}{9y^8}\\cdot \\dfrac{24y^4}{55x^7}\\text{ reduces to }\\dfrac{5}{3y^4}\\cdot \\dfrac{8}{11x^5}, \\text{ which equals }\\dfrac{40}{33x^5y^4}\\]\r\n\r\n(25 and 55 reduce to 5 and 11, 24 and 9 reduce to 8 and 3, x2<\/sup> and x7<\/sup> reduce to x5<\/sup>, y4<\/sup> and y8<\/sup> reduce to y4<\/sup>)\r\n\r\n<\/div>\r\n<\/div>\r\n

Remember: when dividing fractions, reciprocate the dividing fraction.<\/span><\/p>\r\n\r\n

\r\n

Example 8.2.4<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nReduce and divide \\(\\dfrac{a^4b^2}{a}\\) by \\(\\dfrac{b^4}{4}\\).\r\n

\\(\\dfrac{a^4b^2}{a} \\div \\dfrac{b^4}{4}\\text{ reciprocates to } \\dfrac{a^4b^2}{a}\\cdot \\dfrac{4}{b^4}, \\text{ which reduces to }\\dfrac{a^3}{1}\\cdot \\dfrac{4}{b^2}, \\text{ which equals }\\dfrac{4a^3}{b^2}\\)<\/p>\r\n(After reciprocating, 4a4<\/sup>b2<\/sup> and b4<\/sup> reduce to 4a3<\/sup> and b2<\/sup>)\r\n\r\n<\/div>\r\n<\/div>\r\n

In dividing or multiplying some fractions, the polynomials in the fractions must be factored first.<\/span><\/p>\r\n\r\n

\r\n

Example 8.2.5<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nReduce, factor and multiply \\(\\dfrac{x^2-9}{x^2+x-20}\\) and \\(\\dfrac{x^2-8x+16}{3x+9}\\).\r\n\r\n\\[\\dfrac{x^2-9}{x^2+x-20}\\cdot \\dfrac{x^2-8x+16}{3x+9}\\text{ factors to }\\dfrac{(x+3)(x-3)}{(x-4)(x+5)}\\cdot \\dfrac{(x-4)(x-4)}{3(x+3)}\\]\r\n\r\nDividing or cancelling out the common factors \\((x + 3)\\) and \\((x - 4)\\) leaves us with \\(\\dfrac{x-3}{x+5}\\cdot \\dfrac{x-4}{3}\\), which results in \\(\\dfrac{(x-3)(x-4)}{3(x+5)}\\).\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 8.2.6<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nReduce, factor and multiply or divide the following fractions:\r\n\r\n\\[\\dfrac{a^2+7a+10}{a^2+6a+5}\\cdot \\dfrac{a+1}{a^2+4a+4}\\div \\dfrac{a-1}{a+2}\\]\r\n\r\nFactoring each fraction and reciprocating the last one yields:\r\n\r\n\\[\\dfrac{(a+5)(a+2)}{(a+5)(a+1)}\\cdot \\dfrac{(a+1)}{(a+2)(a+2)}\\cdot \\dfrac{(a+2)}{(a-1)}\\]\r\n\r\nDividing or cancelling out the common polynomials leaves us with:\r\n\r\n\\[\\dfrac{1}{a-1}\\]\r\n\r\n<\/div>\r\n<\/div>\r\n

Questions<\/h1>\r\n

Simplify each expression.<\/p>\r\n\r\n

    \r\n \t
  1. \\(\\dfrac{8x^2}{9}\\cdot \\dfrac{9}{2}\\)<\/li>\r\n \t
  2. \\(\\dfrac{8x}{3}\\div \\dfrac{4x}{7}\\)<\/li>\r\n \t
  3. \\(\\dfrac{5x^2}{4}\\cdot \\dfrac{6}{5}\\)<\/li>\r\n \t
  4. \\(\\dfrac{10p}{5}\\div \\dfrac{8}{10}\\)<\/li>\r\n \t
  5. \\(\\dfrac{(m-6)}{7(7m-5)}\\cdot \\dfrac{5m(7m-5)}{m-6}\\)<\/li>\r\n \t
  6. \\(\\dfrac{7(n-2)}{10(n+3)}\\div \\dfrac{n-2}{(n+3)}\\)<\/li>\r\n \t
  7. \\(\\dfrac{7r}{7r(r+10)}\\div \\dfrac{r-6}{(r-6)^2}\\)<\/li>\r\n \t
  8. \\(\\dfrac{6x(x+4)}{(x-3)}\\cdot \\dfrac{(x-3)(x-6)}{6x(x-6)}\\)<\/li>\r\n \t
  9. \\(\\dfrac{x-10}{35x+21}\\div \\dfrac{7}{35x+21}\\)<\/li>\r\n \t
  10. \\(\\dfrac{v-1}{4}\\cdot \\dfrac{4}{v^2-11v+10}\\)<\/li>\r\n \t
  11. \\(\\dfrac{x^2-6x-7}{x+5}\\cdot \\dfrac{x+5}{x-7}\\)<\/li>\r\n \t
  12. \\(\\dfrac{1}{a-6}\\cdot \\dfrac{8a+80}{8}\\)<\/li>\r\n \t
  13. \\(\\dfrac{4m+36}{m+9}\\cdot \\dfrac{m-5}{5m^2}\\)<\/li>\r\n \t
  14. \\(\\dfrac{2r}{r+6}\\div \\dfrac{2r}{7r+42}\\)<\/li>\r\n \t
  15. \\(\\dfrac{n-7}{6n-12}\\cdot \\dfrac{12-6n}{n^2-13n+42}\\)<\/li>\r\n \t
  16. \\(\\dfrac{x^2+11x+24}{6x^3+18x^2}\\cdot \\dfrac{6x^3+6x^2}{x^2+5x-24}\\)<\/li>\r\n \t
  17. \\(\\dfrac{27a+36}{9a+63}\\div \\dfrac{6a+8}{2}\\)<\/li>\r\n \t
  18. \\(\\dfrac{k-7}{k^2-k-12}\\cdot \\dfrac{7k^2-28k}{8k^2-56k}\\)<\/li>\r\n \t
  19. \\(\\dfrac{x^2-12x+32}{x^2-6x-16}\\cdot \\dfrac{7x^2+14x}{7x^2+21x}\\)<\/li>\r\n \t
  20. \\(\\dfrac{9x^3+54x^2}{x^2+5x-14}\\cdot \\dfrac{x^2+5x-14}{10x^2}\\)<\/li>\r\n \t
  21. \\((10m^2+100m)\\cdot \\dfrac{18m^3-36m^2}{20m^2-40m}\\)<\/li>\r\n \t
  22. \\(\\dfrac{n-7}{n^2-2n-35}\\div \\dfrac{9n+54}{10n+50}\\)<\/li>\r\n \t
  23. \\(\\dfrac{x^2-1}{2x-4}\\cdot \\dfrac{x^2-4}{x^2-x-2}\\div \\dfrac{x^2+x-2}{3x-6}\\)<\/li>\r\n \t
  24. \\(\\dfrac{a^3+b^3}{a^2+3ab+2b^2}\\cdot \\dfrac{3a-6b}{3a^2-3ab+3b^2}\\div \\dfrac{a^2-4b^2}{a+2b}\\)<\/li>\r\n<\/ol>\r\nAnswer Key 8.2<\/a>","rendered":"

    Multiplying and dividing rational expressions is very similar to the process used to multiply and divide fractions.<\/p>\n

    \n
    \n

    Example 8.2.1<\/p>\n<\/header>\n

    \n

    Reduce and multiply \"\dfrac{15}{49}\" and \"\dfrac{14}{45}\".<\/p>\n

      <\/span>   <\/span>\"\[\dfrac{15}{49}\cdot \dfrac{14}{45}\text{ reduces to }\dfrac{1}{7}\cdot \dfrac{2}{3}, \text { which equals }\dfrac{2}{21}\]\"<\/p>\n

    (15 and 45 reduce to 1 and 3, and 14 and 49 reduce to 2 and 7)<\/p>\n<\/div>\n<\/div>\n

    This process of multiplication is identical to division, except the first step is to reciprocate any fraction that is being divided. <\/span><\/p>\n

    \n
    \n

    Example 8.2.2<\/p>\n<\/header>\n

    \n

    Reduce and divide \"\dfrac{25}{18}\" by \"\dfrac{15}{6}\".<\/p>\n

    \"\dfrac{25}{18} \div \dfrac{15}{6} \text{ reciprocates to } \dfrac{25}{18}\cdot \dfrac{6}{15}, \text{ which reduces to }\dfrac{5}{3}\cdot \dfrac{1}{3}, \text{ which equals } \dfrac{5}{9}\"<\/p>\n

    (25 and 15 reduce to 5 and 3, and 6 and 18 reduce to 1 and 3)<\/p>\n<\/div>\n<\/div>\n

    When multiplying with rational expressions,\u00a0 follow the same process: first, divide out common factors, then multiply straight across. <\/span><\/p>\n

    \n
    \n

    Example 8.2.3<\/p>\n<\/header>\n

    \n

    Reduce and multiply \"\dfrac{25x^2}{9y^8}\" and \"\dfrac{24y^4}{55x^7}\".<\/p>\n

      <\/span>   <\/span>\"\[\dfrac{25x^2}{9y^8}\cdot \dfrac{24y^4}{55x^7}\text{ reduces to }\dfrac{5}{3y^4}\cdot \dfrac{8}{11x^5}, \text{ which equals }\dfrac{40}{33x^5y^4}\]\"<\/p>\n

    (25 and 55 reduce to 5 and 11, 24 and 9 reduce to 8 and 3, x2<\/sup> and x7<\/sup> reduce to x5<\/sup>, y4<\/sup> and y8<\/sup> reduce to y4<\/sup>)<\/p>\n<\/div>\n<\/div>\n

    Remember: when dividing fractions, reciprocate the dividing fraction.<\/span><\/p>\n

    \n
    \n

    Example 8.2.4<\/p>\n<\/header>\n

    \n

    Reduce and divide \"\dfrac{a^4b^2}{a}\" by \"\dfrac{b^4}{4}\".<\/p>\n

    \"\dfrac{a^4b^2}{a} \div \dfrac{b^4}{4}\text{ reciprocates to } \dfrac{a^4b^2}{a}\cdot \dfrac{4}{b^4}, \text{ which reduces to }\dfrac{a^3}{1}\cdot \dfrac{4}{b^2}, \text{ which equals }\dfrac{4a^3}{b^2}\"<\/p>\n

    (After reciprocating, 4a4<\/sup>b2<\/sup> and b4<\/sup> reduce to 4a3<\/sup> and b2<\/sup>)<\/p>\n<\/div>\n<\/div>\n

    In dividing or multiplying some fractions, the polynomials in the fractions must be factored first.<\/span><\/p>\n

    \n
    \n

    Example 8.2.5<\/p>\n<\/header>\n

    \n

    Reduce, factor and multiply \"\dfrac{x^2-9}{x^2+x-20}\" and \"\dfrac{x^2-8x+16}{3x+9}\".<\/p>\n

      <\/span>   <\/span>\"\[\dfrac{x^2-9}{x^2+x-20}\cdot \dfrac{x^2-8x+16}{3x+9}\text{ factors to }\dfrac{(x+3)(x-3)}{(x-4)(x+5)}\cdot \dfrac{(x-4)(x-4)}{3(x+3)}\]\"<\/p>\n

    Dividing or cancelling out the common factors \"(x + 3)\" and \"(x - 4)\" leaves us with \"\dfrac{x-3}{x+5}\cdot \dfrac{x-4}{3}\", which results in \"\dfrac{(x-3)(x-4)}{3(x+5)}\".<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    Example 8.2.6<\/p>\n<\/header>\n

    \n

    Reduce, factor and multiply or divide the following fractions:<\/p>\n

      <\/span>   <\/span>\"\[\dfrac{a^2+7a+10}{a^2+6a+5}\cdot \dfrac{a+1}{a^2+4a+4}\div \dfrac{a-1}{a+2}\]\"<\/p>\n

    Factoring each fraction and reciprocating the last one yields:<\/p>\n

      <\/span>   <\/span>\"\[\dfrac{(a+5)(a+2)}{(a+5)(a+1)}\cdot \dfrac{(a+1)}{(a+2)(a+2)}\cdot \dfrac{(a+2)}{(a-1)}\]\"<\/p>\n

    Dividing or cancelling out the common polynomials leaves us with:<\/p>\n

      <\/span>   <\/span>\"\[\dfrac{1}{a-1}\]\"<\/p>\n<\/div>\n<\/div>\n

    Questions<\/h1>\n

    Simplify each expression.<\/p>\n

      \n
    1. \"\dfrac{8x^2}{9}\cdot \dfrac{9}{2}\"<\/li>\n
    2. \"\dfrac{8x}{3}\div \dfrac{4x}{7}\"<\/li>\n
    3. \"\dfrac{5x^2}{4}\cdot \dfrac{6}{5}\"<\/li>\n
    4. \"\dfrac{10p}{5}\div \dfrac{8}{10}\"<\/li>\n
    5. \"\dfrac{(m-6)}{7(7m-5)}\cdot \dfrac{5m(7m-5)}{m-6}\"<\/li>\n
    6. \"\dfrac{7(n-2)}{10(n+3)}\div \dfrac{n-2}{(n+3)}\"<\/li>\n
    7. \"\dfrac{7r}{7r(r+10)}\div \dfrac{r-6}{(r-6)^2}\"<\/li>\n
    8. \"\dfrac{6x(x+4)}{(x-3)}\cdot \dfrac{(x-3)(x-6)}{6x(x-6)}\"<\/li>\n
    9. \"\dfrac{x-10}{35x+21}\div \dfrac{7}{35x+21}\"<\/li>\n
    10. \"\dfrac{v-1}{4}\cdot \dfrac{4}{v^2-11v+10}\"<\/li>\n
    11. \"\dfrac{x^2-6x-7}{x+5}\cdot \dfrac{x+5}{x-7}\"<\/li>\n
    12. \"\dfrac{1}{a-6}\cdot \dfrac{8a+80}{8}\"<\/li>\n
    13. \"\dfrac{4m+36}{m+9}\cdot \dfrac{m-5}{5m^2}\"<\/li>\n
    14. \"\dfrac{2r}{r+6}\div \dfrac{2r}{7r+42}\"<\/li>\n
    15. \"\dfrac{n-7}{6n-12}\cdot \dfrac{12-6n}{n^2-13n+42}\"<\/li>\n
    16. \"\dfrac{x^2+11x+24}{6x^3+18x^2}\cdot \dfrac{6x^3+6x^2}{x^2+5x-24}\"<\/li>\n
    17. \"\dfrac{27a+36}{9a+63}\div \dfrac{6a+8}{2}\"<\/li>\n
    18. \"\dfrac{k-7}{k^2-k-12}\cdot \dfrac{7k^2-28k}{8k^2-56k}\"<\/li>\n
    19. \"\dfrac{x^2-12x+32}{x^2-6x-16}\cdot \dfrac{7x^2+14x}{7x^2+21x}\"<\/li>\n
    20. \"\dfrac{9x^3+54x^2}{x^2+5x-14}\cdot \dfrac{x^2+5x-14}{10x^2}\"<\/li>\n
    21. \"(10m^2+100m)\cdot \dfrac{18m^3-36m^2}{20m^2-40m}\"<\/li>\n
    22. \"\dfrac{n-7}{n^2-2n-35}\div \dfrac{9n+54}{10n+50}\"<\/li>\n
    23. \"\dfrac{x^2-1}{2x-4}\cdot \dfrac{x^2-4}{x^2-x-2}\div \dfrac{x^2+x-2}{3x-6}\"<\/li>\n
    24. \"\dfrac{a^3+b^3}{a^2+3ab+2b^2}\cdot \dfrac{3a-6b}{3a^2-3ab+3b^2}\div \dfrac{a^2-4b^2}{a+2b}\"<\/li>\n<\/ol>\n

      Answer Key 8.2<\/a><\/p>\n","protected":false},"author":540,"menu_order":18,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-646","chapter","type-chapter","status-publish","hentry"],"part":386,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/646","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/users\/540"}],"version-history":[{"count":11,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/646\/revisions"}],"predecessor-version":[{"id":3627,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/646\/revisions\/3627"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/parts\/386"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/646\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/media?parent=646"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapter-type?post=646"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/contributor?post=646"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/license?post=646"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}