{"id":9119,"date":"2026-06-01T21:33:48","date_gmt":"2026-06-01T21:33:48","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9119"},"modified":"2026-06-01T21:33:48","modified_gmt":"2026-06-01T21:33:48","slug":"square-roots-decimals","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/square-roots-decimals\/","title":{"rendered":"Square Roots & Decimals"},"content":{"rendered":"

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 √0.25 = 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

      √0.49 = 0.7 because 0.7 × 0.7 = 0.49
      \n√0.0121 = 0.11 because 0.11 × 0.11 = 0.0121
      \n√2 ≈ 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: To find √(decimal), convert the decimal to a fraction, then take the square root of numerator and denominator separately, then convert back.<\/p>\n

      Example 1: √0.09 = √(9\/100) = √9 \/ √100 = 3\/10 = 0.3<\/p>\n

      Example 2: √0.0036 = √(36\/10000) = √36 \/ √10000 = 6\/100 = 0.06<\/p>\n

      Example 3: √1.44 = √(144\/100) = √144 \/ √100 = 12\/10 = 1.2<\/p>\n

      Example 4: √0.0004 = √(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

      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

      Quick Rule:<\/strong> √(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> If 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

      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

      Important:<\/strong> 0.4 has only 1 decimal place, so it cannot be a perfect square decimal (√0.4 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> Find √0.0225<\/p>\n

        0.0225 = 225\/10000 = (15²)\/(100²)
        \n√0.0225 = 15\/100 = 0.15<\/p>\n

        Example 2:<\/strong> Find √0.000625<\/p>\n

        0.000625 = 625\/1000000 = (25²)\/(1000²)
        \n√0.000625 = 25\/1000 = 0.025<\/p>\n

        Example 3:<\/strong> Find √2.25<\/p>\n

        2.25 = 225\/100 = (15²)\/(10²)
        \n√2.25 = 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> Find √0.64<\/p>\n

          Group: . 64
          \nFirst group after decimal: 64
          \nLargest square ≤ 64 is 8² = 64
          \n√0.64 = 0.8<\/p>\n

          Example:<\/strong> Find √0.9 (approx.)<\/p>\n

          Group: . 90 00 00
          \n8² = 64 ≤ 90, remainder 26
          \nBring down 00 → 2600
          \nDouble the quotient (8×2=16). Find digit d such that 16d × d ≤ 2600 → d=1 (161×1=161) → remainder 99
          \nSo √0.9 ≈ 0.948… (actually √0.9 = 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 – Use perfect square decimals as benchmarks<\/strong><\/p>\n

          Example:<\/strong> Estimate √0.3<\/p>\n

          Perfect squares near 0.3: 0.25 (√0.25=0.5) and 0.36 (√0.36=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√0.3 ≈ 0.55 (actual 0.5477…)<\/p>\n

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

          √0.3 = √(3\/10) = √3\/√10 ≈ 1.732\/3.162 ≈ 0.548<\/p>\n

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

          0.55² = 0.3025 (a bit high), so √0.3 ≈ 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
          • √0.25 = 0.5 (0.5 > 0.25)<\/li>\n
          • √0.09 = 0.3 (0.3 > 0.09)<\/li>\n
          • √0.01 = 0.1 (0.1 > 0.01)<\/li>\n<\/ul>\n

            Reason:<\/strong> When 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
            • √1.44 = 1.2 (1.2 < 1.44)<\/li>\n
            • √2.25 = 1.5 (1.5 < 2.25)<\/li>\n
            • √3.24 = 1.8 (1.8 < 3.24)<\/li>\n<\/ul>\n

              Reason:<\/strong> For 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

              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 m² → side = √2.25 = 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 → c=√(0.36+0.64)=√1.0=1.0<\/p>\n<\/td>\n<\/tr>\n

              \n

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

              		\n

              Distance = √[(0.5)² + (1.2)²] = √(0.25+1.44)=√1.69=1.3<\/p>\n<\/td>\n<\/tr>\n

              \n

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

              		\n

              √0.2 ≈ 0.447 for calculations<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

               <\/p>\n

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

              Example 1 – Perfect Square Decimal:<\/strong> Find √0.36<\/p>\n

              Solution:<\/strong> √0.36 = √(36\/100) = 6\/10 = 0.6<\/p>\n

              Answer:<\/strong> 0.6<\/p>\n

               <\/p>\n

              Example 2 – Perfect Square Decimal:<\/strong> Find √0.0025<\/p>\n

              Solution:<\/strong> 0.0025 = 25\/10000 = 5²\/100² → √0.0025 = 5\/100 = 0.05<\/p>\n

              Answer:<\/strong> 0.05<\/p>\n

               <\/p>\n

              Example 3 – Factor Method:<\/strong> Find √0.0144<\/p>\n

              Solution:<\/strong> 0.0144 = 144\/10000 = (12²)\/(100²) = 12\/100 = 0.12<\/p>\n

              Answer:<\/strong> 0.12<\/p>\n

               <\/p>\n

              Example 4 – Greater than 1:<\/strong> Find √2.56<\/p>\n

              Solution:<\/strong> 2.56 = 256\/100 = (16²)\/(10²) → √2.56 = 16\/10 = 1.6<\/p>\n

              Answer:<\/strong> 1.6<\/p>\n

               <\/p>\n

              Example 5 – Estimation:<\/strong> Estimate √0.5<\/p>\n

              Solution:<\/strong> Between √0.49=0.7 and √0.64=0.8, closer to 0.7. √0.5 ≈ 0.707<\/p>\n

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

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

              Mistake 1 – Mismatching decimal places<\/strong>
              \nThinking √0.4 = 0.2 (wrong: 0.2² = 0.04).
              \nCorrect understanding: 0.4 has only 1 decimal place, so its square root cannot be a terminating decimal.<\/p>\n

              Mistake 2 – Ignoring zeros in decimal<\/strong>
              \n0.0004 = 4\/10000, √0.0004 = 2\/100 = 0.02, not 0.2.
              \nCorrect understanding: Count total decimal places carefully.<\/p>\n

              Mistake 3 – Forgetting to convert to fraction<\/strong>
              \nInstead of √0.49 = √49\/√100 = 7\/10 = 0.7, some incorrectly try direct division.
              \nCorrect understanding: Convert to fraction when possible.<\/p>\n

              Mistake 4 – Placing decimal incorrectly in the root<\/strong>
              \n√0.0169 = 0.13 (2 decimal places in original → 1 decimal place in root? Actually 0.13² = 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 → root has n decimal places.<\/p>\n

              Mistake 5 – Thinking all decimals have square roots that are decimals<\/strong>
              \n√0.5 is irrational (≈0.707), not a terminating decimal.
              \nCorrect understanding: Only perfect square decimals have terminating square roots.<\/p>\n

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

               <\/p>\n

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

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

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

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

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

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

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

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

                 <\/p>\n","protected":false},"excerpt":{"rendered":"

                Unit: Squares, Cubes & Roots Chapter: Square Roots & Decimals 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, […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[593],"tags":[],"class_list":["post-9119","post","type-post","status-publish","format-standard","hentry","category-grade-8"],"_links":{"self":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9119","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/comments?post=9119"}],"version-history":[{"count":0,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9119\/revisions"}],"wp:attachment":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/media?parent=9119"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/categories?post=9119"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/tags?post=9119"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}