{"id":690,"date":"2024-04-24T18:47:40","date_gmt":"2024-04-24T18:47:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=690"},"modified":"2024-11-20T00:53:34","modified_gmt":"2024-11-20T00:53:34","slug":"radicals-and-rational-exponents-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/radicals-and-rational-exponents-learn-it-4\/","title":{"raw":"Radicals and Rational Exponents: Learn It 4","rendered":"Radicals and Rational Exponents: Learn It 4"},"content":{"raw":"

Adding and Subtracting Square Roots<\/h2>\r\nTo add or subtract square roots, the expressions involved must have identical radicands and the same type of roots, like square roots. When these conditions are met, you can combine the expressions.\r\n\r\n
For example, consider the expressions [latex]\\sqrt{2}[\/latex] and [latex]3\\sqrt{2}[\/latex]. Since they both contain the square root of 2, they can be combined:
[latex] \\sqrt{2} + 3\\sqrt{2} = (1 + 3)\\sqrt{2} = 4\\sqrt{2} [\/latex]<\/center>Similarly, [latex]3\\sqrt{2} - 2\\sqrt{2}[\/latex] results in [latex](3 - 2)\\sqrt{2} = \\sqrt{2}[\/latex].<\/section>
Notice that simplifying radical expressions is a critical step. The expression [latex]\\sqrt{18}[\/latex] can be simplified to [latex]3\\sqrt{2}[\/latex], because [latex]\\sqrt{18} = \\sqrt{9 \\times 2} = 3\\sqrt{2}[\/latex]. Simplifying each term ensures that you are working with the simplest form, making it easier to add or subtract the radicals.<\/section>
How To: Given a radical expression requiring addition or subtraction of square roots, solve.<\/strong>\r\n
    \r\n \t
  1. Simplify each radical expression:<\/strong> Convert each term to its simplest radical form.<\/li>\r\n \t
  2. Combine terms with matching radicands:<\/strong> Add or subtract the coefficients of the radicals.<\/li>\r\n<\/ol>\r\n<\/section>
    Simplify the expression:\r\n
    [latex]5\\sqrt{12}+2\\sqrt{3}[\/latex]<\/center>\r\n[reveal-answer q=\"413591\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"413591\"][latex]\\begin{align*} 5\\sqrt{12} + 2\\sqrt{3} &= 5\\sqrt{4 \\cdot 3} + 2\\sqrt{3} & \\text{Factor 12 into 4 and 3.} \\\\ &= 5\\cdot 2\\sqrt{3} + 2\\sqrt{3} & \\text{Simplify }\\sqrt{4} \\text{ to } 2. \\\\ &= 10\\sqrt{3} + 2\\sqrt{3} & \\text{Multiply } 5 \\text{ and } 2. \\\\ &= (10 + 2)\\sqrt{3} & \\text{Combine like terms.} \\\\ &= 12\\sqrt{3} & \\text{Finally, simplify the expression.} \\end{align*}[\/latex][\/hidden-answer]<\/section>
    [ohm2_question hide_question_numbers=1]18757[\/ohm2_question]<\/section>
    Simplify the expression:
    [latex]20\\sqrt{72{a}^{3}{b}^{4}c}-14\\sqrt{8{a}^{3}{b}^{4}c}[\/latex]<\/center>[reveal-answer q=\"93063\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"93063\"]\r\nFirst, let's rewrite each term so they have equal radicands.\r\n[latex]\\begin{align*} 20\\sqrt{72a^3b^4c} &= 20\\sqrt{36 \\cdot 2 \\cdot a^3 \\cdot b^4 \\cdot c} & \\text{Factor out the square of 36 from 72.} \\\\ &= 20 \\cdot 6 \\sqrt{2a^3b^4c} & \\text{Simplify } \\sqrt{36} \\text{ to } 6. \\\\ &= 120 \\sqrt{2a^3b^4c} & \\text{Combine constants.}\\\\ 14\\sqrt{8a^3b^4c} &= 14\\sqrt{4 \\cdot 2 \\cdot a^3 \\cdot b^4 \\cdot c} & \\text{Factor out the square of 4 from 8.} \\\\ &= 14 \\cdot 2 \\sqrt{2a^3b^4c} & \\text{Simplify } \\sqrt{4} \\text{ to } 2. \\\\ &= 28 \\sqrt{2a^3b^4c} & \\text{Combine constants.} \\\\\\end{align*}[\/latex]Now the terms have the same radicand so we can subtract.[latex]\\begin{align*}20\\sqrt{72a^3b^4c} - 14\\sqrt{8a^3b^4c} &= 120 \\sqrt{2a^3b^4c} - 28 \\sqrt{2a^3b^4c} & \\text{Subtract the simplified terms.} \\\\ &= (120 - 28) \\sqrt{2a^3b^4c} & \\text{Factor out the common radical term.} \\\\ &= 92 \\sqrt{2a^3b^4c} & \\text{Final expression.} \\end{align*}[\/latex]Do you notice that we can simplify it further?[latex]\\begin{align*} 92 \\sqrt{2a^3b^4c} &= 92 \\sqrt{2 \\cdot a^2 \\cdot a \\cdot b^4 \\cdot c} & \\text{Express } a^3 \\text{ as } a^2 \\cdot a. \\\\ &= 92 \\cdot \\sqrt{a^2} \\cdot \\sqrt{b^4} \\cdot \\sqrt{2ac} & \\text{Separate terms under the radical.} \\\\ &= 92 \\cdot |a| \\cdot b^2 \\cdot \\sqrt{2ac} & \\text{Simplify } \\sqrt{a^2} \\text{ to } |a| \\text{ and } \\sqrt{b^4} \\text{ to } b^2. \\\\ &= 92|a|b^2 \\sqrt{2ac} & \\text{Combine the simplified terms.} \\end{align*}[\/latex]Thus, [latex]20\\sqrt{72a^3b^4c} - 14\\sqrt{8a^3b^4c}= 92|a|b^2 \\sqrt{2ac} [\/latex].\r\n\r\n
    \r\n\r\nA different way of simplifying:\r\n\r\nWe can simplify each square root first to its most simplified term.\r\n\r\n[latex]\\begin{align*} 20\\sqrt{72a^3b^4c} &= 20\\sqrt{36 \\cdot 2 \\cdot a^2 \\cdot a \\cdot b^4 \\cdot c} & \\text{Factor \\(72\\) and \\(a^3\\) into squares and other parts.} \\\\ &= 20\\sqrt{36} \\cdot \\sqrt{2} \\cdot \\sqrt{a^2} \\cdot \\sqrt{a} \\cdot \\sqrt{b^4} \\cdot \\sqrt{c} & \\text{Separate each term under the square root.} \\\\ &= 20 \\cdot 6 \\cdot \\sqrt{2} \\cdot |a| \\cdot b^2 \\cdot \\sqrt{a} \\cdot \\sqrt{c} & \\text{Simplify each square root, where } \\sqrt{b^4} = b^2 \\text{ and } \\sqrt{a^2} = |a|. \\\\ &= 120|a|b^2 \\sqrt{2ac} & \\text{Combine constants and simplify terms.} \\\\ 14\\sqrt{8a^3b^4c} &= 14\\sqrt{4 \\cdot 2 \\cdot a^2 \\cdot a \\cdot b^4 \\cdot c} & \\text{Factor \\(8\\) and \\(a^3\\) similarly.} \\\\ &= 14\\sqrt{4} \\cdot \\sqrt{2} \\cdot \\sqrt{a^2} \\cdot \\sqrt{a} \\cdot \\sqrt{b^4} \\cdot \\sqrt{c} & \\text{Separate terms similarly.} \\\\ &= 14 \\cdot 2 \\cdot \\sqrt{2} \\cdot |a| \\cdot b^2 \\cdot \\sqrt{a} \\cdot \\sqrt{c} & \\text{Simplify.} \\\\ &= 28|a|b^2 \\sqrt{2ac} & \\text{Combine constants and terms.} \\\\\\end{align*}[\/latex]\r\n\r\nNow we can subtract:\r\n\r\n[latex]\\begin{align*}20\\sqrt{72a^3b^4c} - 14\\sqrt{8a^3b^4c} &= 120|a|b^2 \\sqrt{2ac} - 28|a|b^2 \\sqrt{2ac} & \\text{Now subtract the simplified expressions.} \\\\ &= (120 - 28)|a|b^2 \\sqrt{2ac} & \\text{Factor out the common terms.} \\\\ &= 92|a|b^2 \\sqrt{2ac} & \\text{Final simplified expression.} \\end{align*}[\/latex][\/hidden-answer]\r\n\r\n<\/section>
    [ohm2_question hide_question_numbers=1]18758[\/ohm2_question]<\/section>\r\n

    Rationalizing Denominators<\/h3>\r\nWhen an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called \"rationalizing the denominator.\"\r\n\r\n
    Recall the identity property of Multiplication<\/strong>We leverage an important and useful identity in this section in a technique commonly used in college algebra:\r\n

    rewriting an expression by multiplying it by a well-chosen form of the number 1.<\/em><\/p>\r\nBecause the multiplicative identity states that\u00a0[latex]a\\cdot1=a[\/latex],\u00a0we are able to multiply the top and bottom of any fraction by the same number without changing its value. We use this idea when we\u00a0rationalize the denominator.<\/em>\r\n\r\n<\/section>How to Rationalize the Denominator?<\/strong>\r\n\r\nTo rationalize a denominator that contains a square root (or any radical), you multiply both the numerator and the denominator of the fraction by a suitable expression that will eliminate the radical in the denominator.\r\n

      \r\n \t
    • Single Term Denominator:<\/strong> For a single radical term like [latex]b\\sqrt{c}[\/latex] in the denominator, multiply both numerator and denominator by [latex]\\sqrt{c}[\/latex] to eliminate the radical.<\/li>\r\n \t
    • Complex Denominator:<\/strong> If the denominator includes both rational and irrational terms, such as [latex]a + b\\sqrt{c}[\/latex], multiply the fraction by the conjugate [latex]a - b\\sqrt{c}[\/latex] to eliminate the radical. This changes the sign of the radical portion of the denominator.<\/li>\r\n<\/ul>\r\n
      The process of rationalizing a denominator that includes a sum or difference of a rational and an irrational term (like a square root or another radical) involves using the conjugate<\/strong> of the denominator. The conjugate is simply the same terms with the opposite sign between them.If the denominator is [latex]a+\\sqrt{b}[\/latex], then the conjugate is [latex]a-\\sqrt{b}[\/latex]. Multiplying by the conjugate effectively eliminates the radical from the denominator.\r\n\r\n\r\n\r\n\r\n\r\n
      Term<\/strong><\/td>\r\nConjugate<\/strong><\/td>\r\nProduct<\/strong><\/td>\r\n<\/tr>\r\n
      [latex]a+\\sqrt{b}[\/latex]<\/td>\r\n[latex]a-\\sqrt{b}[\/latex]<\/td>\r\n[latex](a+\\sqrt{b})(a-\\sqrt{b}) = (a)^2 - (\\sqrt{b})^2 = a^2-b[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]\\sqrt{2}-3[\/latex]<\/td>\r\n[latex]\\sqrt{2}+3[\/latex]<\/td>\r\n[latex](\\sqrt{2}-3)(\\sqrt{2}+3) = (\\sqrt{2})^2 - (3)^2 = 2-9 = -7[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>
      To rationalize the denominator of [latex]\\dfrac{1}{\\sqrt{a}}[\/latex]:[latex]\\begin{align*} \\frac{1}{\\sqrt{a}} &= \\frac{1}{\\sqrt{a}} \\times \\frac{\\sqrt{a}}{\\sqrt{a}} & \\text{Multiply by } \\frac{\\sqrt{a}}{\\sqrt{a}} \\text{ to rationalize the denominator.} \\\\ &= \\frac{\\sqrt{a}}{\\sqrt{a} \\times \\sqrt{a}} & \\text{Multiply the numerators and the denominators.} \\\\ &= \\frac{\\sqrt{a}}{a} & \\text{Simplify } \\sqrt{a} \\times \\sqrt{a} \\text{ to } a. \\end{align*}[\/latex]<\/section>
      Write [latex]\\dfrac{2\\sqrt{3}}{3\\sqrt{10}}[\/latex] in simplest form.[reveal-answer q=\"982148\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"982148\"]The radical in the denominator is [latex]\\sqrt{10}[\/latex]. So multiply the fraction by [latex]\\dfrac{\\sqrt{10}}{\\sqrt{10}}[\/latex]. Then simplify.\r\n
      [latex]\\begin{align}\\frac{2\\sqrt{3}}{3\\sqrt{10}}\\cdot \\frac{\\sqrt{10}}{\\sqrt{10}} &= \\frac{2\\sqrt{30}}{30} \\\\ &= \\frac{\\sqrt{30}}{15}\\end{align}[\/latex]<\/div>\r\n
      [\/hidden-answer]<\/div>\r\n<\/section>
      [ohm2_question hide_question_numbers=1]18759[\/ohm2_question]<\/section>
      Rationalize the denominator and simplify the expression:
      [latex]\\dfrac{4}{1+\\sqrt{5}}[\/latex]<\/center>[reveal-answer q=\"673394\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"673394\"][latex]\\begin{align*} \\frac{4}{1+\\sqrt{5}} &= \\frac{4}{1+\\sqrt{5}} \\times \\frac{1-\\sqrt{5}}{1-\\sqrt{5}} & \\text{Multiply by the conjugate to rationalize the denominator.} \\\\ &= \\frac{4(1-\\sqrt{5})}{(1+\\sqrt{5})(1-\\sqrt{5})} & \\text{Apply the conjugate.} \\\\ &= \\frac{4 - 4\\sqrt{5}}{1^2 - (\\sqrt{5})^2} & \\text{Expand the numerator and apply the difference of squares.} \\\\ &= \\frac{4 - 4\\sqrt{5}}{1 - 5} & \\text{Simplify the squares and calculate the difference.} \\\\ &= \\frac{4 - 4\\sqrt{5}}{-4} & \\text{Combine like terms in the denominator.} \\\\ &= -1 + \\sqrt{5} & \\text{Divide each term in the numerator by -4.} \\end{align*}[\/latex][\/hidden-answer]<\/section>
      [ohm2_question hide_question_numbers=1]18760[\/ohm2_question]<\/section>","rendered":"

      Adding and Subtracting Square Roots<\/h2>\n

      To add or subtract square roots, the expressions involved must have identical radicands and the same type of roots, like square roots. When these conditions are met, you can combine the expressions.<\/p>\n

      For example, consider the expressions [latex]\\sqrt{2}[\/latex] and [latex]3\\sqrt{2}[\/latex]. Since they both contain the square root of 2, they can be combined:<\/p>\n
      [latex]\\sqrt{2} + 3\\sqrt{2} = (1 + 3)\\sqrt{2} = 4\\sqrt{2}[\/latex]<\/div>\n

      Similarly, [latex]3\\sqrt{2} - 2\\sqrt{2}[\/latex] results in [latex](3 - 2)\\sqrt{2} = \\sqrt{2}[\/latex].<\/section>\n

      Notice that simplifying radical expressions is a critical step. The expression [latex]\\sqrt{18}[\/latex] can be simplified to [latex]3\\sqrt{2}[\/latex], because [latex]\\sqrt{18} = \\sqrt{9 \\times 2} = 3\\sqrt{2}[\/latex]. Simplifying each term ensures that you are working with the simplest form, making it easier to add or subtract the radicals.<\/section>\n
      How To: Given a radical expression requiring addition or subtraction of square roots, solve.<\/strong><\/p>\n
        \n
      1. Simplify each radical expression:<\/strong> Convert each term to its simplest radical form.<\/li>\n
      2. Combine terms with matching radicands:<\/strong> Add or subtract the coefficients of the radicals.<\/li>\n<\/ol>\n<\/section>\n
        Simplify the expression:<\/p>\n
        [latex]5\\sqrt{12}+2\\sqrt{3}[\/latex]<\/div>\n