{"id":9895,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9895"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"square-roots-decimals","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/square-roots-decimals\/","title":{"rendered":"Square Roots & Decimals"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Squares, Cubes & Roots<\/strong><\/h2>\n

Chapter: <\/strong>Square Roots & Decimals<\/strong><\/h3>\n

Reference: – quare Roots of Decimal Numbers, Perfect Square Decimals, Finding Square Root of a Decimal by Prime Factorization, Finding Square Root of a Decimal by Division Method, Square Roots of Non-Perfect Square Decimals, Estimating Decimal Square Roots, Number of Decimal Places in Square Root, Real-Life Applications, Solved Examples, Odd-One-Out Problems, Common Mistakes<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • How to Find Square Roots of Decimal Numbers<\/em><\/li>\n
  • How to Identify Perfect Square Decimals<\/em><\/li>\n
  • How to Use Division Method for Decimal Square Roots<\/em><\/li>\n
  • How to Estimate Square Roots of Non-Perfect Square Decimals<\/em><\/li>\n<\/ul>\n

    Introduction to Square Roots & Decimals<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    Just like whole numbers, decimal numbers can also have square roots. A decimal is a perfect square decimal if its square root is a terminating decimal. For example, 0.25 is a perfect square decimal because \u221a0.25 = 0.5. Not all decimals have neat square roots; many are irrational (non-terminating, non-repeating).<\/p>\n

    When we find square roots of decimals, we essentially ask:<\/p>\n

    “What decimal number, when multiplied by itself, gives this decimal?”<\/p>\n

    Understanding square roots of decimals is essential for working with measurements, areas, and scientific calculations.<\/p>\n

    Importance of Square Roots of Decimals<\/u><\/strong><\/p>\n

      \n
    • Used in geometry (areas of squares with decimal side lengths)<\/li>\n
    • Used in physics and engineering measurements<\/li>\n
    • Used in finance (interest calculations)<\/li>\n
    • Essential for solving quadratic equations with decimal coefficients<\/li>\n
    • Helps in estimating square roots without a calculator<\/li>\n<\/ul>\n

      Example<\/strong><\/p>\n

      \u221a0.49 = 0.7 because 0.7 \u00d7 0.7 = 0.49
      \n\u221a0.0121 = 0.11 because 0.11 \u00d7 0.11 = 0.0121
      \n\u221a2 \u2248 1.4142 (non-terminating, irrational)<\/p>\n

      Subtopics<\/u><\/strong><\/p>\n

      1. Square Roots of Perfect Square Decimals<\/strong><\/p>\n

      A decimal is a perfect square decimal if its square root is a terminating decimal (or integer).<\/p>\n

      Rule:\u00a0To find \u221a(decimal), convert the decimal to a fraction, then take the square root of numerator and denominator separately, then convert back.<\/p>\n

      Example 1:\u00a0\u221a0.09 = \u221a(9\/100) = \u221a9 \/ \u221a100 = 3\/10 = 0.3<\/p>\n

      Example 2:\u00a0\u221a0.0036 = \u221a(36\/10000) = \u221a36 \/ \u221a10000 = 6\/100 = 0.06<\/p>\n

      Example 3:\u00a0\u221a1.44 = \u221a(144\/100) = \u221a144 \/ \u221a100 = 12\/10 = 1.2<\/p>\n

      Example 4:\u00a0\u221a0.0004 = \u221a(4\/10000) = 2\/100 = 0.02<\/p>\n

      Pattern to Remember:<\/strong><\/p>\n

      \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
      \n

      Decimal<\/p>\n<\/td>\n

      		\n

      Fraction<\/p>\n<\/td>\n

      		\n

      Square Root<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      0.01<\/p>\n<\/td>\n

      		\n

      1\/100<\/p>\n<\/td>\n

      		\n

      1\/10 = 0.1<\/p>\n<\/td>\n<\/tr>\n

      \n

      0.04<\/p>\n<\/td>\n

      		\n

      4\/100<\/p>\n<\/td>\n

      		\n

      2\/10 = 0.2<\/p>\n<\/td>\n<\/tr>\n

      \n

      0.09<\/p>\n<\/td>\n

      		\n

      9\/100<\/p>\n<\/td>\n

      		\n

      3\/10 = 0.3<\/p>\n<\/td>\n<\/tr>\n

      \n

      0.16<\/p>\n<\/td>\n

      		\n

      16\/100<\/p>\n<\/td>\n

      		\n

      4\/10 = 0.4<\/p>\n<\/td>\n<\/tr>\n

      \n

      0.25<\/p>\n<\/td>\n

      		\n

      25\/100<\/p>\n<\/td>\n

      		\n

      5\/10 = 0.5<\/p>\n<\/td>\n<\/tr>\n

      \n

      0.36<\/p>\n<\/td>\n

      		\n

      36\/100<\/p>\n<\/td>\n

      		\n

      6\/10 = 0.6<\/p>\n<\/td>\n<\/tr>\n

      \n

      0.49<\/p>\n<\/td>\n

      		\n

      49\/100<\/p>\n<\/td>\n

      		\n

      7\/10 = 0.7<\/p>\n<\/td>\n<\/tr>\n

      \n

      0.64<\/p>\n<\/td>\n

      		\n

      64\/100<\/p>\n<\/td>\n

      		\n

      8\/10 = 0.8<\/p>\n<\/td>\n<\/tr>\n

      \n

      0.81<\/p>\n<\/td>\n

      		\n

      81\/100<\/p>\n<\/td>\n

      		\n

      9\/10 = 0.9<\/p>\n<\/td>\n<\/tr>\n

      \n

      1.00<\/p>\n<\/td>\n

      		\n

      100\/100<\/p>\n<\/td>\n

      		\n

      10\/10 = 1.0<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Quick Rule:<\/strong>\u00a0\u221a(0.0a) where a is a perfect square? Careful with decimal places.<\/p>\n

      2. Number of Decimal Places in Square Root<\/strong><\/p>\n

      When a decimal has an even number of decimal places, it may be a perfect square decimal.<\/p>\n

      Rule:<\/strong>\u00a0If a decimal has 2n decimal places and the number formed by the digits is a perfect square, then its square root will have n decimal places.<\/p>\n

      Examples:<\/strong><\/p>\n

      \n\n\n\n\n\n\n\n\n
      \n

      Decimal<\/p>\n<\/td>\n

      		\n

      Decimal Places<\/p>\n<\/td>\n

      		\n

      Square Root<\/p>\n<\/td>\n

      		\n

      Decimal Places in Root<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      0.49<\/p>\n<\/td>\n

      		\n

      2<\/p>\n<\/td>\n

      		\n

      0.7<\/p>\n<\/td>\n

      		\n

      1<\/p>\n<\/td>\n<\/tr>\n

      \n

      0.0121<\/p>\n<\/td>\n

      		\n

      4<\/p>\n<\/td>\n

      		\n

      0.11<\/p>\n<\/td>\n

      		\n

      2<\/p>\n<\/td>\n<\/tr>\n

      \n

      0.000144<\/p>\n<\/td>\n

      		\n

      6<\/p>\n<\/td>\n

      		\n

      0.012<\/p>\n<\/td>\n

      		\n

      3<\/p>\n<\/td>\n<\/tr>\n

      \n

      1.21<\/p>\n<\/td>\n

      		\n

      2<\/p>\n<\/td>\n

      		\n

      1.1<\/p>\n<\/td>\n

      		\n

      1<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Important:<\/strong>\u00a00.4 has only 1 decimal place, so it cannot be a perfect square decimal (\u221a0.4 is irrational).<\/p>\n

      3. Finding Square Root of a Decimal by Prime Factorization Method<\/strong><\/p>\n

      Steps:<\/strong><\/p>\n

        \n
      1. Write the decimal as a fraction in simplest form<\/li>\n
      2. Find the square root of numerator and denominator separately<\/li>\n
      3. Convert back to decimal<\/li>\n<\/ol>\n

        Example 1:<\/strong>\u00a0Find \u221a0.0225<\/p>\n

        0.0225 = 225\/10000 = (15\u00b2)\/(100\u00b2)
        \n\u221a0.0225 = 15\/100 = 0.15<\/p>\n

        Example 2:<\/strong>\u00a0Find \u221a0.000625<\/p>\n

        0.000625 = 625\/1000000 = (25\u00b2)\/(1000\u00b2)
        \n\u221a0.000625 = 25\/1000 = 0.025<\/p>\n

        Example 3:<\/strong>\u00a0Find \u221a2.25<\/p>\n

        2.25 = 225\/100 = (15\u00b2)\/(10\u00b2)
        \n\u221a2.25 = 15\/10 = 1.5<\/p>\n

        4. Finding Square Root of a Decimal by Division Method<\/strong><\/p>\n

        The long division method for square roots works for decimals as well. Group digits after the decimal in pairs (00, 00, 00…).<\/p>\n

        Steps:<\/strong><\/p>\n

          \n
        1. Group digits before and after decimal in pairs (from decimal point outward)<\/li>\n
        2. Find the largest number whose square is less than or equal to the first group<\/li>\n
        3. Bring down pairs of zeros after decimal as needed<\/li>\n
        4. Continue the division process to get decimal places<\/li>\n<\/ol>\n

          Example:<\/strong>\u00a0Find \u221a0.64<\/p>\n

          Group: . 64
          \nFirst group after decimal: 64
          \nLargest square \u2264 64 is 8\u00b2 = 64
          \n\u221a0.64 = 0.8<\/p>\n

          Example:<\/strong>\u00a0Find \u221a0.9 (approx.)<\/p>\n

          Group: . 90 00 00
          \n8\u00b2 = 64 \u2264 90, remainder 26
          \nBring down 00 \u2192 2600
          \nDouble the quotient (8\u00d72=16). Find digit d such that 16d \u00d7 d \u2264 2600 \u2192 d=1 (161\u00d71=161) \u2192 remainder 99
          \nSo \u221a0.9 \u2248 0.948… (actually \u221a0.9 = 0.94868…)<\/p>\n

          5. Square Roots of Non-Perfect Square Decimals<\/strong><\/p>\n

          Most decimals are not perfect squares. Their square roots are irrational (non-terminating, non-repeating). We can estimate them.<\/p>\n

          Estimating Square Roots of Decimals:<\/strong><\/p>\n

          Method 1 \u2013 Use perfect square decimals as benchmarks<\/strong><\/p>\n

          Example:<\/strong>\u00a0Estimate \u221a0.3<\/p>\n

          Perfect squares near 0.3: 0.25 (\u221a0.25=0.5) and 0.36 (\u221a0.36=0.6)
          \n0.3 is closer to 0.25? Actually 0.3 – 0.25 = 0.05, 0.36 – 0.3 = 0.06, so slightly closer to 0.25
          \n\u221a0.3 \u2248 0.55 (actual 0.5477…)<\/p>\n

          Method 2 \u2013 Convert to fraction and estimate<\/strong><\/p>\n

          \u221a0.3 = \u221a(3\/10) = \u221a3\/\u221a10 \u2248 1.732\/3.162 \u2248 0.548<\/p>\n

          Method 3 \u2013 Use calculator (or approximate decimal multiplication)<\/strong><\/p>\n

          0.55\u00b2 = 0.3025 (a bit high), so \u221a0.3 \u2248 0.547<\/p>\n

          6. Square Roots of Decimals Between 0 and 1<\/strong><\/p>\n

          For decimals between 0 and 1, the square root is larger than the original number.<\/p>\n

          Examples:<\/strong><\/p>\n

            \n
          • \u221a0.25 = 0.5 (0.5 > 0.25)<\/li>\n
          • \u221a0.09 = 0.3 (0.3 > 0.09)<\/li>\n
          • \u221a0.01 = 0.1 (0.1 > 0.01)<\/li>\n<\/ul>\n

            Reason:<\/strong>\u00a0When you multiply a number less than 1 by itself, you get an even smaller number.<\/p>\n

            7. Square Roots of Decimals Greater than 1<\/strong><\/p>\n

            For decimals greater than 1 (but not whole numbers), the square root is smaller than the original number.<\/p>\n

            Examples:<\/strong><\/p>\n

              \n
            • \u221a1.44 = 1.2 (1.2 < 1.44)<\/li>\n
            • \u221a2.25 = 1.5 (1.5 < 2.25)<\/li>\n
            • \u221a3.24 = 1.8 (1.8 < 3.24)<\/li>\n<\/ul>\n

              Reason:<\/strong>\u00a0For numbers greater than 1, squaring makes them larger.<\/p>\n

              8. Real-Life Applications of Decimal Square Roots<\/strong><\/p>\n

              \n\n\n\n\n\n\n\n\n
              \n

              Application<\/p>\n<\/td>\n

              		\n

              Example<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

              \n

              Finding side length from area (in decimal units)<\/p>\n<\/td>\n

              		\n

              Area = 2.25 m\u00b2 \u2192 side = \u221a2.25 = 1.5 m<\/p>\n<\/td>\n<\/tr>\n

              \n

              Pythagorean theorem with decimal sides<\/p>\n<\/td>\n

              		\n

              a=0.6, b=0.8 \u2192 c=\u221a(0.36+0.64)=\u221a1.0=1.0<\/p>\n<\/td>\n<\/tr>\n

              \n

              Distance calculations with coordinates<\/p>\n<\/td>\n

              		\n

              Distance = \u221a[(0.5)\u00b2 + (1.2)\u00b2] = \u221a(0.25+1.44)=\u221a1.69=1.3<\/p>\n<\/td>\n<\/tr>\n

              \n

              Physics (velocity, acceleration)<\/p>\n<\/td>\n

              		\n

              \u221a0.2 \u2248 0.447 for calculations<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              Source: Kapdec.com<\/p>\n<\/div>\n

              \u00a0<\/p>\n

              Solved Examples<\/strong><\/p>\n

              Example 1 \u2013 Perfect Square Decimal:<\/strong>\u00a0Find \u221a0.36<\/p>\n

              Solution:<\/strong>\u00a0\u221a0.36 = \u221a(36\/100) = 6\/10 = 0.6<\/p>\n

              Answer:<\/strong>\u00a00.6<\/p>\n

              \u00a0<\/p>\n

              Example 2 \u2013 Perfect Square Decimal:<\/strong>\u00a0Find \u221a0.0025<\/p>\n

              Solution:<\/strong>\u00a00.0025 = 25\/10000 = 5\u00b2\/100\u00b2 \u2192 \u221a0.0025 = 5\/100 = 0.05<\/p>\n

              Answer:<\/strong>\u00a00.05<\/p>\n

              \u00a0<\/p>\n

              Example 3 \u2013 Factor Method:<\/strong>\u00a0Find \u221a0.0144<\/p>\n

              Solution:<\/strong>\u00a00.0144 = 144\/10000 = (12\u00b2)\/(100\u00b2) = 12\/100 = 0.12<\/p>\n

              Answer:<\/strong>\u00a00.12<\/p>\n

              \u00a0<\/p>\n

              Example 4 \u2013 Greater than 1:<\/strong>\u00a0Find \u221a2.56<\/p>\n

              Solution:<\/strong>\u00a02.56 = 256\/100 = (16\u00b2)\/(10\u00b2) \u2192 \u221a2.56 = 16\/10 = 1.6<\/p>\n

              Answer:<\/strong>\u00a01.6<\/p>\n

              \u00a0<\/p>\n

              Example 5 \u2013 Estimation:<\/strong>\u00a0Estimate \u221a0.5<\/p>\n

              Solution:<\/strong>\u00a0Between \u221a0.49=0.7 and \u221a0.64=0.8, closer to 0.7. \u221a0.5 \u2248 0.707<\/p>\n

              Answer:<\/strong>\u00a0About 0.707<\/p>\n

              Common Mistakes to Avoid<\/u><\/strong><\/p>\n

              Mistake 1 \u2013 Mismatching decimal places<\/strong>
              \nThinking \u221a0.4 = 0.2 (wrong: 0.2\u00b2 = 0.04).
              \nCorrect understanding: 0.4 has only 1 decimal place, so its square root cannot be a terminating decimal.<\/p>\n

              Mistake 2 \u2013 Ignoring zeros in decimal<\/strong>
              \n0.0004 = 4\/10000, \u221a0.0004 = 2\/100 = 0.02, not 0.2.
              \nCorrect understanding: Count total decimal places carefully.<\/p>\n

              Mistake 3 \u2013 Forgetting to convert to fraction<\/strong>
              \nInstead of \u221a0.49 = \u221a49\/\u221a100 = 7\/10 = 0.7, some incorrectly try direct division.
              \nCorrect understanding: Convert to fraction when possible.<\/p>\n

              Mistake 4 \u2013 Placing decimal incorrectly in the root<\/strong>
              \n\u221a0.0169 = 0.13 (2 decimal places in original \u2192 1 decimal place in root? Actually 0.13\u00b2 = 0.0169, so 2 decimals in original, 2 in root? Wait: 0.13 has 2 decimals, 0.0169 has 4 decimals.
              \nCorrect understanding: Original has 2n decimal places \u2192 root has n decimal places.<\/p>\n

              Mistake 5 \u2013 Thinking all decimals have square roots that are decimals<\/strong>
              \n\u221a0.5 is irrational (\u22480.707), not a terminating decimal.
              \nCorrect understanding: Only perfect square decimals have terminating square roots.<\/p>\n

              Mistake 6 \u2013 Confusing \u221a0.1 with 0.1<\/strong>
              \n\u221a0.1 \u2248 0.316, not 0.1.
              \nCorrect understanding: Square root of a decimal between 0 and 1 is larger than the decimal itself.<\/p>\n

              \u00a0<\/p>\n

              Quick Reference Summary<\/u><\/strong><\/p>\n

              Perfect Square Decimal:<\/strong>\u00a0A decimal with an even number of decimal places whose square root is a terminating decimal<\/p>\n

              Square Root of Decimal (Fraction Method):<\/strong>\u00a0Write decimal as fraction, take \u221a of numerator and denominator<\/p>\n

              Number of Decimal Places:<\/strong>\u00a0If decimal has 2n decimal places and is a perfect square, its square root has n decimal places<\/p>\n

              \u221a of decimals between 0 and 1:<\/strong>\u00a0Result is larger than the original number<\/p>\n

              \u221a of decimals greater than 1:<\/strong>\u00a0Result is smaller than the original number<\/p>\n

              Common Perfect Square Decimals:<\/strong><\/p>\n

                \n
              • 0.01 \u2192 0.1<\/li>\n
              • 0.04 \u2192 0.2<\/li>\n
              • 0.09 \u2192 0.3<\/li>\n
              • 0.16 \u2192 0.4<\/li>\n
              • 0.25 \u2192 0.5<\/li>\n
              • 0.36 \u2192 0.6<\/li>\n
              • 0.49 \u2192 0.7<\/li>\n
              • 0.64 \u2192 0.8<\/li>\n
              • 0.81 \u2192 0.9<\/li>\n
              • 1.00 \u2192 1.0<\/li>\n
              • 1.21 \u2192 1.1<\/li>\n
              • 1.44 \u2192 1.2<\/li>\n<\/ul>\n

                \u00a0<\/p>\n

                <\/p>\n