{"id":3090,"date":"2024-08-28T18:57:36","date_gmt":"2024-08-28T18:57:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=3090"},"modified":"2024-11-20T02:44:12","modified_gmt":"2024-11-20T02:44:12","slug":"rational-expressions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/rational-expressions-learn-it-4\/","title":{"raw":"Rational Expressions: Learn It 4","rendered":"Rational Expressions: Learn It 4"},"content":{"raw":"

Simplifying Complex Rational Expressions<\/h2>\r\nImagine a fraction within a fraction\u2014sounds complicated, right? That\u2019s exactly what a complex rational expression looks like! It could have fractions in the numerator, the denominator, or even both. But don't worry, simplifying them isn't as tricky as it might seem.\r\n\r\nTo make these expressions easier to work with, your goal is to combine everything in the numerator into one single fraction, and do the same for the denominator. Once you have one fraction over another, you can simplify the expression just like you would divide any two fractions: by multiplying the top fraction by the reciprocal (or the flip) of the bottom fraction.\r\n\r\n
How To: Given a complex rational expression, simplify it<\/strong>\r\n
    \r\n \t
  1. Combine the expressions in the numerator into a single rational expression by adding or subtracting.<\/li>\r\n \t
  2. Combine the expressions in the denominator into a single rational expression by adding or subtracting.<\/li>\r\n \t
  3. Rewrite as the numerator divided by the denominator.<\/li>\r\n \t
  4. Rewrite as multiplication.<\/li>\r\n \t
  5. Multiply.<\/li>\r\n \t
  6. Simplify.<\/li>\r\n<\/ol>\r\n<\/section>
    Simplify: [latex]\\dfrac{a}{\\dfrac{1}{b}+c}[\/latex]Solution\r\n<\/strong>\r\nStep 1: Rewrite the Numerator<\/strong>\r\nThe numerator [latex]a[\/latex] can be expressed as [latex]\\dfrac{a}{1}[\/latex] to facilitate operations.\r\n\r\nStep 2: Combine Expressions in the Denominator<\/strong>\r\nThe denominator contains [latex]\\dfrac{1}{b}+c[\/latex]. Find a common denominator, which is [latex]b[\/latex], resulting in: [latex]\\dfrac{1+bc}{b}[\/latex].\r\n\r\nStep 3: Rewrite as Multiplication<\/strong>\r\nUse the reciprocal of the combined denominator to convert the division into multiplication:\r\n

    [latex]\\dfrac{a}{1}\\cdot \\dfrac{b}{1+bc}[\/latex]<\/p>\r\nStep 4: Simplify<\/strong>\r\nThe multiplication of the fractions simplifies to:\r\n

    [latex]\\dfrac{ab}{1+bc}[\/latex]<\/p>\r\n\r\n<\/section>

    Can a complex rational expression always be simplified?<\/strong>Yes. We can always rewrite a complex rational expression as a simplified rational expression.<\/em><\/section>
    Simplify: [latex]\\dfrac{y+\\dfrac{1}{x}}{\\dfrac{x}{y}}[\/latex] .[reveal-answer q=\"967019\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"967019\"]Begin by combining the expressions in the numerator into one expression.\r\n
    [latex]\\begin{array}{cc}y\\cdot \\dfrac{x}{x}+\\dfrac{1}{x}\\hfill & \\text{Multiply by }\\dfrac{x}{x}\\text{to get LCD as denominator}.\\hfill \\\\ \\dfrac{xy}{x}+\\dfrac{1}{x}\\hfill & \\\\ \\dfrac{xy+1}{x}\\hfill & \\text{Add numerators}.\\hfill \\end{array}[\/latex]<\/div>\r\nNow the numerator is a single rational expression and the denominator is a single rational expression.\r\n
    [latex]\\dfrac{\\dfrac{xy+1}{x}}{\\dfrac{x}{y}}[\/latex]<\/div>\r\nWe can rewrite this as division and then multiplication.\r\n
    [latex]\\begin{array}{cc}\\dfrac{xy+1}{x}\\div \\dfrac{x}{y}\\hfill & \\\\ \\dfrac{xy+1}{x}\\cdot \\dfrac{y}{x}\\hfill & \\text{Rewrite as multiplication}\\text{.}\\hfill \\\\ \\dfrac{y\\left(xy+1\\right)}{{x}^{2}}\\hfill & \\text{Multiply}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\r\n
    [\/hidden-answer]<\/div>\r\n<\/section>
    [ohm2_question hide_question_numbers=1]18906[\/ohm2_question]<\/section>
    [ohm2_question hide_question_numbers=1]18907[\/ohm2_question]<\/section>
    [ohm2_question hide_question_numbers=1]18908[\/ohm2_question]<\/section>","rendered":"

    Simplifying Complex Rational Expressions<\/h2>\n

    Imagine a fraction within a fraction\u2014sounds complicated, right? That\u2019s exactly what a complex rational expression looks like! It could have fractions in the numerator, the denominator, or even both. But don’t worry, simplifying them isn’t as tricky as it might seem.<\/p>\n

    To make these expressions easier to work with, your goal is to combine everything in the numerator into one single fraction, and do the same for the denominator. Once you have one fraction over another, you can simplify the expression just like you would divide any two fractions: by multiplying the top fraction by the reciprocal (or the flip) of the bottom fraction.<\/p>\n

    How To: Given a complex rational expression, simplify it<\/strong><\/p>\n
      \n
    1. Combine the expressions in the numerator into a single rational expression by adding or subtracting.<\/li>\n
    2. Combine the expressions in the denominator into a single rational expression by adding or subtracting.<\/li>\n
    3. Rewrite as the numerator divided by the denominator.<\/li>\n
    4. Rewrite as multiplication.<\/li>\n
    5. Multiply.<\/li>\n
    6. Simplify.<\/li>\n<\/ol>\n<\/section>\n
      Simplify: [latex]\\dfrac{a}{\\dfrac{1}{b}+c}[\/latex]Solution
      \n<\/strong>
      \nStep 1: Rewrite the Numerator<\/strong>
      \nThe numerator [latex]a[\/latex] can be expressed as [latex]\\dfrac{a}{1}[\/latex] to facilitate operations.
      \n
      \nStep 2: Combine Expressions in the Denominator<\/strong>
      \nThe denominator contains [latex]\\dfrac{1}{b}+c[\/latex]. Find a common denominator, which is [latex]b[\/latex], resulting in: [latex]\\dfrac{1+bc}{b}[\/latex].
      \n
      \nStep 3: Rewrite as Multiplication<\/strong>
      \nUse the reciprocal of the combined denominator to convert the division into multiplication:<\/p>\n

      [latex]\\dfrac{a}{1}\\cdot \\dfrac{b}{1+bc}[\/latex]<\/p>\n

      Step 4: Simplify<\/strong>
      \nThe multiplication of the fractions simplifies to:<\/p>\n

      [latex]\\dfrac{ab}{1+bc}[\/latex]<\/p>\n<\/section>\n

      Can a complex rational expression always be simplified?<\/strong>Yes. We can always rewrite a complex rational expression as a simplified rational expression.<\/em><\/section>\n
      Simplify: [latex]\\dfrac{y+\\dfrac{1}{x}}{\\dfrac{x}{y}}[\/latex] .<\/p>\n