{"id":262,"date":"2015-09-18T20:23:13","date_gmt":"2015-09-18T20:23:13","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=262"},"modified":"2015-11-03T00:13:19","modified_gmt":"2015-11-03T00:13:19","slug":"evaluating-square-roots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/evaluating-square-roots\/","title":{"raw":"Evaluating Square Roots","rendered":"Evaluating Square Roots"},"content":{"raw":"When the square root of a number is squared, the result is the original number. Since [latex]{4}^{2}=16[\/latex], the square root of [latex]16[\/latex] is [latex]4[\/latex]. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.\r\n\r\nIn general terms, if [latex]a[\/latex] is a positive real number, then the square root of [latex]a[\/latex] is a number that, when multiplied by itself, gives [latex]a[\/latex]. The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root<\/strong> is the nonnegative number that when multiplied by itself equals [latex]a[\/latex]. The square root obtained using a calculator is the principal square root.\r\n\r\nThe principal square root of [latex]a[\/latex] is written as [latex]\\sqrt{a}[\/latex]. The symbol is called a radical<\/strong>, the term under the symbol is called the radicand<\/strong>, and the entire expression is called a radical expression<\/strong>.\r\n\r\n\"The\r\n
\r\n

A General Note: Principal Square Root<\/h3>\r\nThe principal square root<\/strong> of [latex]a[\/latex] is the nonnegative number that, when multiplied by itself, equals [latex]a[\/latex]. It is written as a radical expression<\/strong>, with a symbol called a radical<\/strong> over the term called the radicand<\/strong>: [latex]\\sqrt{a}[\/latex].\r\n\r\n<\/div>\r\n
\r\n

Q & A<\/h3>\r\n

Does<\/strong> [latex]\\sqrt{25}=\\pm 5[\/latex]?<\/h3>\r\nNo. Although both<\/em> [latex]{5}^{2}[\/latex] and<\/em> [latex]{\\left(-5\\right)}^{2}[\/latex] are<\/em> [latex]25[\/latex], the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is<\/em> [latex]\\sqrt{25}=5[\/latex].\r\n\r\n<\/div>\r\n
\r\n

Example 1: Evaluating Square Roots<\/h3>\r\nEvaluate each expression.\r\n
    \r\n\t
  1. [latex]\\sqrt{100}[\/latex]<\/li>\r\n\t
  2. [latex]\\sqrt{\\sqrt{16}}[\/latex]<\/li>\r\n\t
  3. [latex]\\sqrt{25+144}[\/latex]<\/li>\r\n\t
  4. [latex]\\sqrt{49}-\\sqrt{81}\\\\[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\n
      \r\n\t
    1. [latex]\\sqrt{100}=10[\/latex] because [latex]{10}^{2}=100[\/latex]<\/li>\r\n\t
    2. [latex]\\sqrt{\\sqrt{16}}=\\sqrt{4}=2[\/latex] because [latex]{4}^{2}=16[\/latex] and [latex]{2}^{2}=4[\/latex]<\/li>\r\n\t
    3. [latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]{13}^{2}=169[\/latex]<\/li>\r\n\t
    4. [latex]\\sqrt{49}-\\sqrt{81}=7 - 9=-2[\/latex] because [latex]{7}^{2}=49[\/latex] and [latex]{9}^{2}=81[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Q & A<\/h3>\r\n

      For [latex]\\sqrt{25+144}[\/latex], can we find the square roots before adding?<\/h3>\r\nNo.<\/em> [latex]\\sqrt{25}+\\sqrt{144}=5+12=17[\/latex]. This is not equivalent to<\/em> [latex]\\sqrt{25+144}=13[\/latex]. The order of operations requires us to add the terms in the radicand before finding the square root.<\/em>\r\n\r\n<\/div>\r\n
      \r\n

      Try It 1<\/h3>\r\nEvaluate each expression.\r\n

      a. [latex]\\sqrt{225}[\/latex]\r\nb. [latex]\\sqrt{\\sqrt{81}}[\/latex]\r\nc. [latex]\\sqrt{25 - 9}[\/latex]\r\nd. [latex]\\sqrt{36}+\\sqrt{121}[\/latex]<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

      When the square root of a number is squared, the result is the original number. Since [latex]{4}^{2}=16[\/latex], the square root of [latex]16[\/latex] is [latex]4[\/latex]. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.<\/p>\n

      In general terms, if [latex]a[\/latex] is a positive real number, then the square root of [latex]a[\/latex] is a number that, when multiplied by itself, gives [latex]a[\/latex]. The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root<\/strong> is the nonnegative number that when multiplied by itself equals [latex]a[\/latex]. The square root obtained using a calculator is the principal square root.<\/p>\n

      The principal square root of [latex]a[\/latex] is written as [latex]\\sqrt{a}[\/latex]. The symbol is called a radical<\/strong>, the term under the symbol is called the radicand<\/strong>, and the entire expression is called a radical expression<\/strong>.<\/p>\n

      \"The<\/p>\n

      \n

      A General Note: Principal Square Root<\/h3>\n

      The principal square root<\/strong> of [latex]a[\/latex] is the nonnegative number that, when multiplied by itself, equals [latex]a[\/latex]. It is written as a radical expression<\/strong>, with a symbol called a radical<\/strong> over the term called the radicand<\/strong>: [latex]\\sqrt{a}[\/latex].<\/p>\n<\/div>\n

      \n

      Q & A<\/h3>\n

      Does<\/strong> [latex]\\sqrt{25}=\\pm 5[\/latex]?<\/h3>\n

      No. Although both<\/em> [latex]{5}^{2}[\/latex] and<\/em> [latex]{\\left(-5\\right)}^{2}[\/latex] are<\/em> [latex]25[\/latex], the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is<\/em> [latex]\\sqrt{25}=5[\/latex].<\/p>\n<\/div>\n

      \n

      Example 1: Evaluating Square Roots<\/h3>\n

      Evaluate each expression.<\/p>\n

        \n
      1. [latex]\\sqrt{100}[\/latex]<\/li>\n
      2. [latex]\\sqrt{\\sqrt{16}}[\/latex]<\/li>\n
      3. [latex]\\sqrt{25+144}[\/latex]<\/li>\n
      4. [latex]\\sqrt{49}-\\sqrt{81}\\\\[\/latex]<\/li>\n<\/ol>\n<\/div>\n
        \n

        Solution<\/h3>\n
          \n
        1. [latex]\\sqrt{100}=10[\/latex] because [latex]{10}^{2}=100[\/latex]<\/li>\n
        2. [latex]\\sqrt{\\sqrt{16}}=\\sqrt{4}=2[\/latex] because [latex]{4}^{2}=16[\/latex] and [latex]{2}^{2}=4[\/latex]<\/li>\n
        3. [latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]{13}^{2}=169[\/latex]<\/li>\n
        4. [latex]\\sqrt{49}-\\sqrt{81}=7 - 9=-2[\/latex] because [latex]{7}^{2}=49[\/latex] and [latex]{9}^{2}=81[\/latex]<\/li>\n<\/ol>\n<\/div>\n
          \n

          Q & A<\/h3>\n

          For [latex]\\sqrt{25+144}[\/latex], can we find the square roots before adding?<\/h3>\n

          No.<\/em> [latex]\\sqrt{25}+\\sqrt{144}=5+12=17[\/latex]. This is not equivalent to<\/em> [latex]\\sqrt{25+144}=13[\/latex]. The order of operations requires us to add the terms in the radicand before finding the square root.<\/em><\/p>\n<\/div>\n

          \n

          Try It 1<\/h3>\n

          Evaluate each expression.<\/p>\n

          a. [latex]\\sqrt{225}[\/latex]
          \nb. [latex]\\sqrt{\\sqrt{81}}[\/latex]
          \nc. [latex]\\sqrt{25 - 9}[\/latex]
          \nd. [latex]\\sqrt{36}+\\sqrt{121}[\/latex]<\/p>\n

          Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

          \n\t\t\t

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