{"id":8627,"date":"2023-10-04T14:39:00","date_gmt":"2023-10-04T14:39:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=8627"},"modified":"2024-10-18T20:56:43","modified_gmt":"2024-10-18T20:56:43","slug":"irrational-numbers-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/irrational-numbers-learn-it-2\/","title":{"raw":"Irrational Numbers: Learn It 2","rendered":"Irrational Numbers: Learn It 2"},"content":{"raw":"

Square Roots for Non-Perfect Square Numbers<\/h2>\r\n

One collection of irrational numbers is square roots<\/strong> of numbers that aren\u2019t perfect squares<\/strong>.\u00a0Recall that, [latex]x[\/latex] is the square root of the number [latex]a[\/latex], denoted [latex]\\sqrt{a}[\/latex], if [latex]x^2=a[\/latex]. The number [latex]a[\/latex] is the perfect square of the integer [latex]n[\/latex] if [latex]a=n^2[\/latex].<\/p>\r\n

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\r\n

perfect square<\/h3>\r\n

Perfect squares are integers that result from squaring a whole number.<\/p>\r\n<\/div>\r\n<\/section>\r\n

\r\n

Square Root Notation<\/strong><\/p>\r\n

[latex]\\sqrt{m}[\/latex] is read as \"the square root of [latex]m\\text{.\"}[\/latex]<\/center>
If [latex]m={n}^{2}[\/latex] then [latex]\\sqrt{m}=n[\/latex] for [latex]{n}\\ge 0[\/latex].<\/center>
\"A<\/center><\/section>\r\n
\r\n

The rational number [latex]\\frac{a}{b}[\/latex] is a perfect square if both [latex]a[\/latex] and [latex]b[\/latex] are perfect squares.<\/p>\r\n<\/section>\r\n

One method of determining if an integer is a perfect square is to examine its prime factorization. If, in that factorization, all the prime factors are raised to even powers, the integer is a perfect square.<\/p>\r\n

Another method is to attempt to factor the integer into an integer squared. It is possible that you recognize the number as a perfect square (such as [latex]4[\/latex] or [latex]9[\/latex]). Or, if you have a calculator at hand, use the calculator to determine if the square root of the integer is an integer.<\/p>\r\n

\r\n

Determine which of the following are perfect squares.<\/p>\r\n

    \r\n\t
  1. [latex]45[\/latex]<\/li>\r\n\t
  2. [latex]81[\/latex]<\/li>\r\n\t
  3. [latex]\\frac{9}{28}[\/latex]<\/li>\r\n\t
  4. [latex]\\frac{144}{400}[\/latex]<\/li>\r\n<\/ol>\r\n

    [reveal-answer q=\"214538\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214538\"]<\/p>\r\n

      \r\n\t
    1. The prime factorization of [latex]45[\/latex] is [latex]45=3^2\u00d75[\/latex]. Since the [latex]5[\/latex] is not raised to an even power, [latex]45[\/latex] is not a perfect square.<\/li>\r\n\t
    2. The prime factorization of [latex]81[\/latex] is [latex]3^4[\/latex]. All the prime factors are raised to even powers, so [latex]81[\/latex] is a perfect square.<\/li>\r\n\t
    3. We must determine if both the numerator and denominator of [latex]\\frac{9}{28}[\/latex] are perfect squares for the rational number to be a perfect square. The numerator is [latex]9[\/latex], and as mentioned above, [latex]9[\/latex] is a perfect square (it is [latex]3[\/latex] squared). Now we check the prime factorization of the denominator, [latex]28[\/latex], which is [latex]28=2^2\u00d77[\/latex]. Since [latex]7[\/latex] is not raised to an even power, [latex]28[\/latex] is not a perfect square. Since the denominator is not a perfect square, [latex]\\frac{9}{28}[\/latex] is not a perfect square.<\/li>\r\n\t
    4. We must determine if both the numerator and denominator of [latex]\\frac{144}{400}[\/latex] are perfect squares for the rational number to be a perfect square. The numerator is [latex]144[\/latex]. The prime factorization of [latex]144[\/latex] is [latex]144=2^4\u00d73^2[\/latex]. Since all the prime factors of [latex]144[\/latex] are raised to even powers, [latex]144[\/latex] is a perfect square. Now we check the prime factorization of the denominator, [latex]400[\/latex], which is [latex]400=2^4\u00d75^2[\/latex]. Since all the prime factors of [latex]400[\/latex] are raised to even powers, [latex]400[\/latex] is a perfect square. Since the numerator and denominator of [latex]\\frac{144}{400}[\/latex] are perfect squares, [latex]\\frac{144}{400}[\/latex] is a perfect square.<\/li>\r\n<\/ol>\r\n

      [\/hidden-answer]<\/p>\r\n<\/section>\r\n

      \r\n

      Using Desmos to Determine if a Number Is a Perfect Square<\/p>\r\n<\/center>\r\n

      Desmos may be used to determine if a number is a perfect square by using its square root function. When Desmos is opened, there is a tab in the lower left-hand corner of the Desmos screen. This tab opens the Desmos keypad, shown below.<\/p>\r\n

      \"Desmos<\/center>\r\n

       <\/p>\r\n\r\nThere you find the key for the square root, which is circled in the image above. To find the square root of a number, click the square root key, which begins a calculation, and then enter the value for which you want a square root. If the result is an integer, then the number is a perfect square. For more information on this, watch this video on Using Desmos to Find the Square Root of a Number.<\/a>\r\n

      You can view the transcript for \u201cSquare Roots on Desmos Calculator\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n

      Square roots of perfect squares are always whole numbers, so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.<\/p>\r\n

      \r\n
      \r\n

      square roots of non perfect squares<\/h3>\r\n

      The decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.<\/p>\r\n<\/div>\r\n<\/section>\r\n

      Identify each of the following as rational or irrational:\r\n\r\n
        \r\n\t
      1. [latex]\\sqrt{36}[\/latex]<\/li>\r\n\t
      2. [latex]\\sqrt{44}[\/latex]<\/li>\r\n<\/ol>\r\n\r\n[reveal-answer q=\"237122\"]Show Solution[\/reveal-answer] [hidden-answer a=\"237122\"]\r\n\r\n
          \r\n\t
        1. The number [latex]36[\/latex] is a perfect square, since [latex]{6}^{2}=36[\/latex]. So [latex]\\sqrt{36}=6[\/latex]. Therefore [latex]\\sqrt{36}[\/latex] is rational.<\/li>\r\n\t
        2. Remember that [latex]{6}^{2}=36[\/latex] and [latex]{7}^{2}=49[\/latex], so [latex]44[\/latex] is not a perfect square. This means [latex]\\sqrt{44}[\/latex] is irrational.<\/li>\r\n<\/ol>\r\n\r\n[\/hidden-answer]<\/section>\r\n
          \r\n

          [ohm2_question hide_question_numbers=1]12703[\/ohm2_question]<\/p>\r\n<\/section>","rendered":"

          Square Roots for Non-Perfect Square Numbers<\/h2>\n

          One collection of irrational numbers is square roots<\/strong> of numbers that aren\u2019t perfect squares<\/strong>.\u00a0Recall that, [latex]x[\/latex] is the square root of the number [latex]a[\/latex], denoted [latex]\\sqrt{a}[\/latex], if [latex]x^2=a[\/latex]. The number [latex]a[\/latex] is the perfect square of the integer [latex]n[\/latex] if [latex]a=n^2[\/latex].<\/p>\n

          \n
          \n

          perfect square<\/h3>\n

          Perfect squares are integers that result from squaring a whole number.<\/p>\n<\/div>\n<\/section>\n

          \n

          Square Root Notation<\/strong><\/p>\n

          [latex]\\sqrt{m}[\/latex] is read as “the square root of [latex]m\\text{.\"}[\/latex]<\/div>\n
          If [latex]m={n}^{2}[\/latex] then [latex]\\sqrt{m}=n[\/latex] for [latex]{n}\\ge 0[\/latex].<\/div>\n
          \"A<\/div>\n<\/section>\n
          \n

          The rational number [latex]\\frac{a}{b}[\/latex] is a perfect square if both [latex]a[\/latex] and [latex]b[\/latex] are perfect squares.<\/p>\n<\/section>\n

          One method of determining if an integer is a perfect square is to examine its prime factorization. If, in that factorization, all the prime factors are raised to even powers, the integer is a perfect square.<\/p>\n

          Another method is to attempt to factor the integer into an integer squared. It is possible that you recognize the number as a perfect square (such as [latex]4[\/latex] or [latex]9[\/latex]). Or, if you have a calculator at hand, use the calculator to determine if the square root of the integer is an integer.<\/p>\n

          \n

          Determine which of the following are perfect squares.<\/p>\n

            \n
          1. [latex]45[\/latex]<\/li>\n
          2. [latex]81[\/latex]<\/li>\n
          3. [latex]\\frac{9}{28}[\/latex]<\/li>\n
          4. [latex]\\frac{144}{400}[\/latex]<\/li>\n<\/ol>\n