{"id":711,"date":"2024-04-24T21:14:52","date_gmt":"2024-04-24T21:14:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=711"},"modified":"2025-08-21T23:02:36","modified_gmt":"2025-08-21T23:02:36","slug":"radicals-and-rational-exponents-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/radicals-and-rational-exponents-learn-it-5\/","title":{"raw":"Radicals and Rational Exponents: Learn It 5","rendered":"Radicals and Rational Exponents: Learn It 5"},"content":{"raw":"

Rational Roots<\/h2>\r\nAlthough square roots are the most commonly recognized, we can also find cube roots, fourth roots, fifth roots, and beyond. Just as the square root function reverses the squaring function, these higher-order roots are the inverses of their corresponding power functions. These root functions are particularly useful when you need to identify the number that, when raised to a specific power, results in a given number.\r\n\r\n
\r\n

[latex]\\text{n}^{\\text{th}}[\/latex] Root<\/h3>\r\nThe [latex]\\text{n}^{\\text{th}}[\/latex] root of [latex]a[\/latex], denoted as [latex]\\sqrt[n]{a}[\/latex], is a number that, when raised to the nth power, equals [latex]a[\/latex].\r\n\r\n<\/section>
For example, if you are looking for the cube root ([latex]3^{\\text{rd}}[\/latex] root) of [latex]8[\/latex], you need to find a number that, when multiplied by itself three times, gives [latex]8[\/latex]. In this case, [latex]2 \\cdot 2 \\cdot 2 = 8[\/latex], so the cube root of [latex]8[\/latex] is [latex]2[\/latex]. That is, [latex]\\sqrt[3]{8} = 2[\/latex].<\/section>Now that we've explored [latex]\\text{n}^{\\text{th}}[\/latex] roots, let's look into a specific type of [latex]\\text{n}^{\\text{th}}[\/latex] root known as the principal [latex]\\text{n}^{\\text{th}}[\/latex] root<\/strong>. When we talk about the principal [latex]\\text{n}^{\\text{th}}[\/latex] root, we're focusing on the most common or main root that we use in math, especially when the root needs to be positive.\r\n\r\n
\r\n

principal [latex]\\text{n}^{\\text{th}}[\/latex] root<\/h3>\r\nIf [latex]a[\/latex] is a real number with at least one [latex]\\text{n}^{\\text{th}}[\/latex] root, then the principal [latex]\\text{n}^{\\text{th}}[\/latex] root<\/strong> of [latex]a[\/latex], written as [latex]\\sqrt[n]{a}[\/latex], is the number with the same sign as [latex]a[\/latex] that, when raised to the [latex]\\text{n}^{\\text{th}}[\/latex] power, equals [latex]a[\/latex].\r\n\r\n<\/section>
Here, [latex]n[\/latex] is what we call the index<\/strong> of the radical, which tells us the degree of the root.<\/section>
Simplify each of the following:\r\n
    \r\n \t
  1. [latex]\\sqrt[5]{-32}[\/latex]<\/li>\r\n \t
  2. [latex]\\sqrt[4]{4}\\cdot \\sqrt[4]{1,024}[\/latex]<\/li>\r\n \t
  3. [latex]-\\sqrt[3]{\\dfrac{8{x}^{6}}{125}}[\/latex]<\/li>\r\n \t
  4. [latex]8\\sqrt[4]{3}-\\sqrt[4]{48}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"579304\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"579304\"]\r\n
      \r\n \t
    1. [latex]\\sqrt[5]{-32}=-2[\/latex] because [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\r\n \t
    2. First, express the product as a single radical expression. [latex]\\sqrt[4]{4\\text{,}096}=8[\/latex] because [latex]{8}^{4}=4,096[\/latex]<\/li>\r\n \t
    3. [latex]\\begin{align} -\\sqrt[3]{\\frac{8x^6}{125}} &= \\frac{-\\sqrt[3]{8x^6}}{\\sqrt[3]{125}} && \\text{Write as quotient of two radical expressions.} \\\\ &= \\frac{-2x^2}{5} && \\text{Simplify.} \\end{align}[\/latex]<\/li>\r\n \t
    4. [latex]\\begin{align}\\\\ 8\\sqrt[4]{3}-\\sqrt[4]{48} & = 8\\sqrt[4]{3}-2\\sqrt[4]{3} && \\text{Simplify to get equal radicands}. \\\\ & = 6\\sqrt[4]{3} && \\text{Add}. \\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
      [ohm2_question hide_question_numbers=1]18761[\/ohm2_question]<\/section>
      [ohm2_question hide_question_numbers=1]18762[\/ohm2_question]<\/section>\r\n

      Rational Exponents<\/h2>\r\nRadical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[\/latex] is even, then [latex]x[\/latex] cannot be negative.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Radical Form<\/td>\r\nExponent Form<\/td>\r\n<\/tr>\r\n
      [latex]\\sqrt{x}[\/latex]<\/span><\/td>\r\n[latex]x^{\\frac{1}{2}}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n
      [latex]\\sqrt[3]{x}[\/latex]<\/span><\/td>\r\n[latex]x^{\\frac{1}{3}}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n
      [latex]\\sqrt[4]{x}[\/latex]<\/span><\/td>\r\n[latex]x^{\\frac{1}{4}}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n
      ...<\/span><\/td>\r\n...<\/span><\/td>\r\n<\/tr>\r\n
      [latex]\\sqrt[n]{x}[\/latex]<\/span><\/td>\r\n[latex]x^{\\frac{1}{n}}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
      We can also have rational exponents with numerators other than 1.<\/section>
      \r\n

      Rational Exponents<\/h3>\r\nRational exponents are another way to express principal [latex]\\text{n}^{\\text{th}}[\/latex] roots.\r\n\r\n \r\n\r\nThe general form for converting between a radical expression with a radical symbol and one with a rational exponent is\r\n
      [latex]\\begin{align}{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}\\end{align}[\/latex]<\/div>\r\n\r\n[caption id=\"attachment_741\" align=\"aligncenter\" width=\"300\"]\"\" A radical with an exponent can be rewritten as a fractional exponent[\/caption]\r\n\r\n<\/section>
      Write [latex]{343}^{\\frac{2}{3}}[\/latex] as a radical and then simplify.\r\n<\/strong>[reveal-answer q=\"755495\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"755495\"][latex]{343}^{\\frac{2}{3}} = \\sqrt[3]{343^2} [\/latex]<\/span> or [latex]{343}^{\\frac{2}{3}} = (\\sqrt[3]{343})^2 [\/latex]<\/span><\/span>To simplify the radical, we know that [latex]343 = 7^3[\/latex]. This means: [latex]343^{\\frac{2}{3}} = \\left(\\sqrt[3]{343}\\right)^2 = \\left(\\sqrt[3]{7^3}\\right)^2 = 7^2 = 49[\/latex][\/hidden-answer]<\/section>
      [ohm2_question hide_question_numbers=1]18763[\/ohm2_question]<\/section>
      Write [latex]\\dfrac{4}{\\sqrt[7]{{a}^{2}}}[\/latex] using a rational exponent.\r\n<\/strong>[reveal-answer q=\"796457\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"796457\"][latex]\\begin{align*} \\text{Start with:} & \\quad \\frac{4}{\\sqrt[7]{a^2}} & \\text{Original expression.} \\\\ \\text{Rewrite the radical:} & \\quad \\frac{4}{a^{\\frac{2}{7}}} & \\text{Express the 7th root of } a^2 \\text{ as } a^{2\/7}. \\\\ \\text{Apply the negative exponent rule:} & \\quad 4a^{-\\frac{2}{7}} & \\text{Transform } \\frac{1}{a^{\\frac{2}{7}}} \\text{ to } a^{-\\frac{2}{7}} \\text{ using the rule } \\frac{1}{a^x} = a^{-x}. \\end{align*}[\/latex]<\/span>[\/hidden-answer]<\/section>
      [ohm2_question hide_question_numbers=1]18764[\/ohm2_question]<\/section>
      Simplify:\r\n
        \r\n \t
      1. [latex]5\\left(2{x}^{\\frac{3}{4}}\\right)\\left(3{x}^{\\frac{1}{5}}\\right)[\/latex]<\/li>\r\n \t
      2. [latex]{\\left(\\dfrac{16}{9}\\right)}^{-\\frac{1}{2}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"803060\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"803060\"]\r\n
          \r\n \t
        1. [latex]\\begin{align}5\\left(2{x}^{\\frac{3}{4}}\\right)\\left(3{x}^{\\frac{1}{5}}\\right) & = 30{x}^{\\frac{3}{4}}{x}^{\\frac{1}{5}}&& \\text{Multiply the coefficients}. \\\\ & = 30{x}^{\\frac{3}{4}+\\frac{1}{5}}&& \\text{Use properties of exponents}. \\\\ & = 30{x}^{\\frac{19}{20}}&& \\text{Simplify}. \\end{align}[\/latex]<\/li>\r\n \t
        2. [latex]\\begin{align}{\\left(\\dfrac{16}{9}\\right)}^{-\\frac{1}{2}} & = {\\left(\\frac{9}{16}\\right)}^{\\frac{1}{2}}&& \\text{Use definition of negative exponents}. \\\\ & = \\sqrt{\\frac{9}{16}}&& \\text{Rewrite as a radical}. \\\\ & = \\frac{\\sqrt{9}}{\\sqrt{16}}&& \\text{Use the quotient rule}. \\\\ & = \\frac{3}{4}&& \\text{Simplify}. \\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
          [ohm2_question hide_question_numbers=1]18765[\/ohm2_question]<\/section>","rendered":"

          Rational Roots<\/h2>\n

          Although square roots are the most commonly recognized, we can also find cube roots, fourth roots, fifth roots, and beyond. Just as the square root function reverses the squaring function, these higher-order roots are the inverses of their corresponding power functions. These root functions are particularly useful when you need to identify the number that, when raised to a specific power, results in a given number.<\/p>\n

          \n

          [latex]\\text{n}^{\\text{th}}[\/latex] Root<\/h3>\n

          The [latex]\\text{n}^{\\text{th}}[\/latex] root of [latex]a[\/latex], denoted as [latex]\\sqrt[n]{a}[\/latex], is a number that, when raised to the nth power, equals [latex]a[\/latex].<\/p>\n<\/section>\n

          For example, if you are looking for the cube root ([latex]3^{\\text{rd}}[\/latex] root) of [latex]8[\/latex], you need to find a number that, when multiplied by itself three times, gives [latex]8[\/latex]. In this case, [latex]2 \\cdot 2 \\cdot 2 = 8[\/latex], so the cube root of [latex]8[\/latex] is [latex]2[\/latex]. That is, [latex]\\sqrt[3]{8} = 2[\/latex].<\/section>\n

          Now that we’ve explored [latex]\\text{n}^{\\text{th}}[\/latex] roots, let’s look into a specific type of [latex]\\text{n}^{\\text{th}}[\/latex] root known as the principal [latex]\\text{n}^{\\text{th}}[\/latex] root<\/strong>. When we talk about the principal [latex]\\text{n}^{\\text{th}}[\/latex] root, we’re focusing on the most common or main root that we use in math, especially when the root needs to be positive.<\/p>\n

          \n

          principal [latex]\\text{n}^{\\text{th}}[\/latex] root<\/h3>\n

          If [latex]a[\/latex] is a real number with at least one [latex]\\text{n}^{\\text{th}}[\/latex] root, then the principal [latex]\\text{n}^{\\text{th}}[\/latex] root<\/strong> of [latex]a[\/latex], written as [latex]\\sqrt[n]{a}[\/latex], is the number with the same sign as [latex]a[\/latex] that, when raised to the [latex]\\text{n}^{\\text{th}}[\/latex] power, equals [latex]a[\/latex].<\/p>\n<\/section>\n

          Here, [latex]n[\/latex] is what we call the index<\/strong> of the radical, which tells us the degree of the root.<\/section>\n
          Simplify each of the following:<\/p>\n
            \n
          1. [latex]\\sqrt[5]{-32}[\/latex]<\/li>\n
          2. [latex]\\sqrt[4]{4}\\cdot \\sqrt[4]{1,024}[\/latex]<\/li>\n
          3. [latex]-\\sqrt[3]{\\dfrac{8{x}^{6}}{125}}[\/latex]<\/li>\n
          4. [latex]8\\sqrt[4]{3}-\\sqrt[4]{48}[\/latex]<\/li>\n<\/ol>\n