{"id":1408,"date":"2021-12-02T19:37:52","date_gmt":"2021-12-03T00:37:52","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/8-2-multiplication-and-division-of-rational-expressions\/"},"modified":"2023-08-30T19:31:03","modified_gmt":"2023-08-30T23:31:03","slug":"multiplication-and-division-of-rational-expressions","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/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":"Multiplying 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 [latex]\\dfrac{15}{49}[\/latex] and [latex]\\dfrac{14}{45}[\/latex].\r\n\r\n[latex]\\dfrac{15}{49}\\cdot \\dfrac{14}{45}\\text{ reduces to }\\dfrac{1}{7}\\cdot \\dfrac{2}{3}, \\text { which equals }\\dfrac{2}{21}[\/latex]\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 [latex]\\dfrac{25}{18}[\/latex] by [latex]\\dfrac{15}{6}[\/latex].\r\n

[latex]\\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}[\/latex]<\/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, 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 [latex]\\dfrac{25x^2}{9y^8}[\/latex] and [latex]\\dfrac{24y^4}{55x^7}[\/latex].\r\n\r\n[latex]\\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}[\/latex]\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 [latex]\\dfrac{a^4b^2}{a}[\/latex] by [latex]\\dfrac{b^4}{4}[\/latex].\r\n

[latex]\\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}[\/latex]<\/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 [latex]\\dfrac{x^2-9}{x^2+x-20}[\/latex] and [latex]\\dfrac{x^2-8x+16}{3x+9}[\/latex].\r\n\r\n[latex]\\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)}[\/latex]\r\n\r\nDividing or cancelling out the common factors [latex](x + 3)[\/latex] and [latex](x - 4)[\/latex] leaves us with [latex]\\dfrac{x-3}{x+5}\\cdot \\dfrac{x-4}{3}[\/latex], which results in [latex]\\dfrac{(x-3)(x-4)}{3(x+5)}[\/latex].\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[latex]\\dfrac{a^2+7a+10}{a^2+6a+5}\\cdot \\dfrac{a+1}{a^2+4a+4}\\div \\dfrac{a-1}{a+2}[\/latex]\r\n\r\nFactoring each fraction and reciprocating the last one yields:\r\n\r\n[latex]\\dfrac{(a+5)(a+2)}{(a+5)(a+1)}\\cdot \\dfrac{(a+1)}{(a+2)(a+2)}\\cdot \\dfrac{(a+2)}{(a-1)}[\/latex]\r\n\r\nDividing or cancelling out the common polynomials leaves us with:\r\n\r\n[latex]\\dfrac{1}{a-1}[\/latex]\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. [latex]\\dfrac{8x^2}{9}\\cdot \\dfrac{9}{2}[\/latex]<\/li>\r\n \t
  2. [latex]\\dfrac{8x}{3}\\div \\dfrac{4x}{7}[\/latex]<\/li>\r\n \t
  3. [latex]\\dfrac{5x^2}{4}\\cdot \\dfrac{6}{5}[\/latex]<\/li>\r\n \t
  4. [latex]\\dfrac{10p}{5}\\div \\dfrac{8}{10}[\/latex]<\/li>\r\n \t
  5. [latex]\\dfrac{(m-6)}{7(7m-5)}\\cdot \\dfrac{5m(7m-5)}{m-6}[\/latex]<\/li>\r\n \t
  6. [latex]\\dfrac{7(n-2)}{10(n+3)}\\div \\dfrac{n-2}{(n+3)}[\/latex]<\/li>\r\n \t
  7. [latex]\\dfrac{7r}{7r(r+10)}\\div \\dfrac{r-6}{(r-6)^2}[\/latex]<\/li>\r\n \t
  8. [latex]\\dfrac{6x(x+4)}{(x-3)}\\cdot \\dfrac{(x-3)(x-6)}{6x(x-6)}[\/latex]<\/li>\r\n \t
  9. [latex]\\dfrac{x-10}{35x+21}\\div \\dfrac{7}{35x+21}[\/latex]<\/li>\r\n \t
  10. [latex]\\dfrac{v-1}{4}\\cdot \\dfrac{4}{v^2-11v+10}[\/latex]<\/li>\r\n \t
  11. [latex]\\dfrac{x^2-6x-7}{x+5}\\cdot \\dfrac{x+5}{x-7}[\/latex]<\/li>\r\n \t
  12. [latex]\\dfrac{1}{a-6}\\cdot \\dfrac{8a+80}{8}[\/latex]<\/li>\r\n \t
  13. [latex]\\dfrac{4m+36}{m+9}\\cdot \\dfrac{m-5}{5m^2}[\/latex]<\/li>\r\n \t
  14. [latex]\\dfrac{2r}{r+6}\\div \\dfrac{2r}{7r+42}[\/latex]<\/li>\r\n \t
  15. [latex]\\dfrac{n-7}{6n-12}\\cdot \\dfrac{12-6n}{n^2-13n+42}[\/latex]<\/li>\r\n \t
  16. [latex]\\dfrac{x^2+11x+24}{6x^3+18x^2}\\cdot \\dfrac{6x^3+6x^2}{x^2+5x-24}[\/latex]<\/li>\r\n \t
  17. [latex]\\dfrac{27a+36}{9a+63}\\div \\dfrac{6a+8}{2}[\/latex]<\/li>\r\n \t
  18. [latex]\\dfrac{k-7}{k^2-k-12}\\cdot \\dfrac{7k^2-28k}{8k^2-56k}[\/latex]<\/li>\r\n \t
  19. [latex]\\dfrac{x^2-12x+32}{x^2-6x-16}\\cdot \\dfrac{7x^2+14x}{7x^2+21x}[\/latex]<\/li>\r\n \t
  20. [latex]\\dfrac{9x^3+54x^2}{x^2+5x-14}\\cdot \\dfrac{x^2+5x-14}{10x^2}[\/latex]<\/li>\r\n \t
  21. [latex](10m^2+100m)\\cdot \\dfrac{18m^3-36m^2}{20m^2-40m}[\/latex]<\/li>\r\n \t
  22. [latex]\\dfrac{n-7}{n^2-2n-35}\\div \\dfrac{9n+54}{10n+50}[\/latex]<\/li>\r\n \t
  23. [latex]\\dfrac{x^2-1}{2x-4}\\cdot \\dfrac{x^2-4}{x^2-x-2}\\div \\dfrac{x^2+x-2}{3x-6}[\/latex]<\/li>\r\n \t
  24. [latex]\\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}[\/latex]<\/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 [latex]\\dfrac{15}{49}[\/latex] and [latex]\\dfrac{14}{45}[\/latex].<\/p>\n

    [latex]\\dfrac{15}{49}\\cdot \\dfrac{14}{45}\\text{ reduces to }\\dfrac{1}{7}\\cdot \\dfrac{2}{3}, \\text { which equals }\\dfrac{2}{21}[\/latex]<\/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 [latex]\\dfrac{25}{18}[\/latex] by [latex]\\dfrac{15}{6}[\/latex].<\/p>\n

    [latex]\\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}[\/latex]<\/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, 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 [latex]\\dfrac{25x^2}{9y^8}[\/latex] and [latex]\\dfrac{24y^4}{55x^7}[\/latex].<\/p>\n

    [latex]\\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}[\/latex]<\/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 [latex]\\dfrac{a^4b^2}{a}[\/latex] by [latex]\\dfrac{b^4}{4}[\/latex].<\/p>\n

    [latex]\\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}[\/latex]<\/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 [latex]\\dfrac{x^2-9}{x^2+x-20}[\/latex] and [latex]\\dfrac{x^2-8x+16}{3x+9}[\/latex].<\/p>\n

    [latex]\\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)}[\/latex]<\/p>\n

    Dividing or cancelling out the common factors [latex](x + 3)[\/latex] and [latex](x - 4)[\/latex] leaves us with [latex]\\dfrac{x-3}{x+5}\\cdot \\dfrac{x-4}{3}[\/latex], which results in [latex]\\dfrac{(x-3)(x-4)}{3(x+5)}[\/latex].<\/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

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

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

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

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

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

    Questions<\/h1>\n

    Simplify each expression.<\/p>\n

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

      Answer Key 8.2<\/a><\/p>\n","protected":false},"author":90,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1408","chapter","type-chapter","status-publish","hentry","license-cc-by-nc-sa"],"part":1404,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1408","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":3,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1408\/revisions"}],"predecessor-version":[{"id":2135,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1408\/revisions\/2135"}],"part":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/parts\/1404"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1408\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=1408"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapter-type?post=1408"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=1408"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=1408"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}