{"id":1494,"date":"2026-08-24T19:00:00","date_gmt":"2026-08-24T13:30:00","guid":{"rendered":"https:\/\/digitoolkit.in\/blog\/?p=1494"},"modified":"2026-08-24T10:19:30","modified_gmt":"2026-08-24T04:49:30","slug":"how-to-calculate-square-root","status":"publish","type":"post","link":"https:\/\/digitoolkit.in\/blog\/how-to-calculate-square-root\/","title":{"rendered":"How to Calculate Square Root (Step by Step)"},"content":{"rendered":"<div style=\"background:#f2f7fb;border-left:4px solid #2271b1;padding:16px 20px;margin:0 0 24px;border-radius:4px;\">\n<p><strong>Quick Answer:<\/strong> To calculate a square root, find the number that, when multiplied by itself, gives your original number. For perfect squares use prime factorisation (&radic;144 = 12); for other numbers use the long division method taught in NCERT Class 8, or estimate between two known squares. The square root of 2025, for example, is 45 because 45 &times; 45 = 2025.<\/p>\n<p><strong>Key takeaways:<\/strong><\/p>\n<ul>\n<li>A square root of a number N is a value that multiplies by itself to give N.<\/li>\n<li>Perfect squares (4, 9, 16, 25&hellip;) have whole-number square roots.<\/li>\n<li>Prime factorisation works cleanly for perfect squares.<\/li>\n<li>The long division method (NCERT Class 8) handles any number, including decimals.<\/li>\n<li>Estimation between nearby perfect squares gives a fast approximate answer.<\/li>\n<\/ul>\n<\/div>\n<p>Square roots appear everywhere in the Indian school syllabus and in competitive exams like SSC, banking, and JEE, yet many students remember only the calculator button and not the method. Knowing how to calculate a square root by hand builds real number sense and is essential when calculators are not allowed in an exam hall. This step-by-step guide covers the three methods every Indian student learns &mdash; prime factorisation, the long division method, and estimation &mdash; with worked examples that follow the NCERT Class 8 approach.<\/p>\n<h2>What a Square Root Means<\/h2>\n<p>The square root of a number N is the value that, multiplied by itself, produces N. It is written with the radical sign &radic;. So &radic;49 = 7 because 7 &times; 7 = 49, and &radic;81 = 9 because 9 &times; 9 = 81. Every positive number technically has two square roots, one positive and one negative (since (&minus;7) &times; (&minus;7) also equals 49), but in everyday and school contexts we usually mean the positive, or principal, root. If you just want the answer quickly, a <a href=\"https:\/\/digitoolkit.in\/calculators\/square-root-calculator\/\">square root calculator<\/a> gives it instantly, but understanding the methods below is what helps in exams.<\/p>\n<h2>Method 1: Prime Factorisation (for Perfect Squares)<\/h2>\n<p>This is the method NCERT introduces first and it works beautifully for perfect squares. The idea is to break the number into prime factors, pair them up, and take one factor from each pair.<\/p>\n<ol>\n<li><strong>Find the prime factors.<\/strong> For 144: 144 = 2 &times; 2 &times; 2 &times; 2 &times; 3 &times; 3.<\/li>\n<li><strong>Group the primes in pairs.<\/strong> (2 &times; 2) &times; (2 &times; 2) &times; (3 &times; 3).<\/li>\n<li><strong>Take one number from each pair.<\/strong> 2 &times; 2 &times; 3 = 12.<\/li>\n<li><strong>That product is the square root.<\/strong> So &radic;144 = 12.<\/li>\n<\/ol>\n<p>If any prime is left without a partner, the number is not a perfect square, and you should switch to the long division method. Finding those prime factors is itself easier if you are comfortable with a <a href=\"https:\/\/digitoolkit.in\/calculators\/hcf-gcd-calculator\/\">HCF and GCD calculator<\/a> style of breaking numbers down.<\/p>\n<h2>Method 2: The Long Division Method (for Any Number)<\/h2>\n<p>This is the workhorse method from NCERT Class 8, Chapter 6, and it handles non-perfect squares and decimals too. Take &radic;1369 as an example.<\/p>\n<ol>\n<li><strong>Pair the digits from the right.<\/strong> 1369 becomes bars over &ldquo;13&rdquo; and &ldquo;69&rdquo;.<\/li>\n<li><strong>Find the largest square &le; the first group.<\/strong> 3&sup2; = 9 &le; 13, so the first digit of the root is 3; subtract 9 from 13 to get 4.<\/li>\n<li><strong>Bring down the next pair.<\/strong> 4 becomes 469.<\/li>\n<li><strong>Double the current root and find the next digit.<\/strong> Double 3 is 6; we need a digit d so that (60 + d) &times; d &le; 469. Testing d = 7 gives 67 &times; 7 = 469 exactly.<\/li>\n<li><strong>Remainder is zero,<\/strong> so &radic;1369 = 37.<\/li>\n<\/ol>\n<blockquote>\n<p><strong>Key takeaway:<\/strong> The long division method never fails. It works for perfect squares, non-perfect squares, and decimals, which is exactly why NCERT teaches it as the universal method for exams where calculators are banned.<\/p>\n<\/blockquote>\n<h2>Method 3: Estimation Between Perfect Squares<\/h2>\n<p>When you only need an approximate answer, estimation is fastest. To estimate &radic;50, note that 7&sup2; = 49 and 8&sup2; = 64. Since 50 is just above 49, &radic;50 is a little more than 7 &mdash; about 7.07. This mental technique is invaluable in aptitude tests where you must eliminate answer options quickly.<\/p>\n<h2>Worked Examples<\/h2>\n<h3>Example 1: A perfect square, &radic;2025<\/h3>\n<p>Prime factorisation of 2025 gives 3 &times; 3 &times; 3 &times; 3 &times; 5 &times; 5 = (3&times;3)(3&times;3)(5&times;5). Taking one from each pair: 3 &times; 3 &times; 5 = 45. So &radic;2025 = 45, a number worth remembering since 2025 is a recent year often used in exam questions.<\/p>\n<h3>Example 2: A non-perfect square, &radic;200<\/h3>\n<p>200 is between 196 (14&sup2;) and 225 (15&sup2;), so the root is between 14 and 15. Using long division gives approximately 14.14. Estimation alone tells you it is close to 14.1, enough for many multiple-choice questions.<\/p>\n<h3>Example 3: A decimal, &radic;2.25<\/h3>\n<p>Recognise that 1.5 &times; 1.5 = 2.25, so &radic;2.25 = 1.5. Decimals often hide simple perfect squares, so always check whether the number is a familiar square in disguise before doing heavy calculation.<\/p>\n<h2>Perfect Squares Reference Table<\/h2>\n<table>\n<thead>\n<tr>\n<th>Number<\/th>\n<th>Square<\/th>\n<th>Number<\/th>\n<th>Square<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>11<\/td>\n<td>121<\/td>\n<td>16<\/td>\n<td>256<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>144<\/td>\n<td>17<\/td>\n<td>289<\/td>\n<\/tr>\n<tr>\n<td>13<\/td>\n<td>169<\/td>\n<td>18<\/td>\n<td>324<\/td>\n<\/tr>\n<tr>\n<td>14<\/td>\n<td>196<\/td>\n<td>19<\/td>\n<td>361<\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>225<\/td>\n<td>20<\/td>\n<td>400<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Benefits of Learning These Methods<\/h2>\n<p>Mastering square roots by hand pays off directly in Indian examinations, where calculators are usually prohibited in board exams and in tests like SSC, RRB, banking, and JEE Main. Beyond exams, the skill sharpens estimation and number sense that helps in geometry, physics, and statistics. Knowing prime factorisation strengthens your grasp of factors and multiples generally, and the long division method reinforces place value and careful arithmetic. Students who understand these methods rarely freeze when a square root appears in an unfamiliar problem.<\/p>\n<h2>Challenges and Limitations<\/h2>\n<p>The long division method is reliable but slow, and it demands neat, careful work &mdash; a single misplaced digit throws off the whole answer. Prime factorisation only gives clean results for perfect squares and becomes tedious for large numbers with big prime factors. Estimation is fast but only approximate, so it cannot be your final answer when an exact value is required. The practical skill is knowing which method fits the question: factorisation for obvious perfect squares, long division for exact non-perfect roots, and estimation for quick elimination.<\/p>\n<h2>Common Mistakes to Avoid<\/h2>\n<ul>\n<li><strong>Pairing digits from the left.<\/strong> In long division you must pair from the right (the units place), or the whole method fails.<\/li>\n<li><strong>Forgetting the second root.<\/strong> Every positive number has both a positive and negative square root; note which one the question wants.<\/li>\n<li><strong>Leaving an unpaired prime unnoticed.<\/strong> If a prime has no partner, the number is not a perfect square; do not force a whole-number answer.<\/li>\n<li><strong>Misplacing the decimal in decimals.<\/strong> When taking the root of a decimal, pair digits carefully around the decimal point.<\/li>\n<li><strong>Confusing square and square root.<\/strong> Squaring makes a number bigger for values above one; the square root makes it smaller.<\/li>\n<li><strong>Rounding too early in estimation.<\/strong> Keep one extra decimal while estimating so your final rounded answer is accurate.<\/li>\n<\/ul>\n<h2>Best Practices and Expert Recommendations<\/h2>\n<ul>\n<li><strong>Memorise squares up to at least 30.<\/strong> Instant recall of perfect squares speeds up every method.<\/li>\n<li><strong>Choose the method to fit the number.<\/strong> Factorise perfect squares, use long division for exact non-perfect roots, estimate for quick checks.<\/li>\n<li><strong>Write neatly in long division.<\/strong> Line up digits carefully to avoid place-value slips.<\/li>\n<li><strong>Always sanity-check with estimation.<\/strong> Confirm your computed root sits between the right pair of perfect squares.<\/li>\n<li><strong>Practise NCERT exercises repeatedly.<\/strong> The Class 8 problems build the speed exams demand.<\/li>\n<li><strong>Verify with a calculator while practising.<\/strong> Use a tool to confirm your manual answer and learn from any error.<\/li>\n<\/ul>\n<p>When your problem also involves splitting numbers evenly, pairing square roots with a <a href=\"https:\/\/digitoolkit.in\/calculators\/division-calculator\/\">division calculator<\/a> helps you check each step of the long division method.<\/p>\n<div data-dtk-related=\"1\" style=\"background:#f8f9fb;border:1px solid #e2e8f0;border-radius:6px;padding:16px 20px;margin:28px 0;\"><strong>Related tools &amp; guides on DigiToolkit<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/digitoolkit.in\/calculators\/square-root-calculator\/\">Try the free Square Root Calculator &rarr;<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/square-root-formula-explained\/\">Square Root Formula Explained with Examples<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/what-is-square-root-simple-guide\/\">What Is Square Root? A Simple Guide<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/square-root-calculator-free-online-tool\/\">Square Root Calculator: Free Online Tool + Guide<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/square-root-examples-for-beginners\/\">Square Root Examples for Beginners<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/adding-fractions-examples-beginners\/\">Adding Fractions Examples for Beginners<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/adding-fractions-calculator-tool-guide\/\">Adding Fractions Calculator: Free Online Tool + Guide<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/category\/mathematics\/\">More Mathematics guides<\/a><\/li>\n<\/ul>\n<\/div>\n<h2>Frequently Asked Questions<\/h2>\n<p><strong>What is the easiest way to find a square root?<\/strong><br \/>For perfect squares, prime factorisation is easiest: break the number into primes, pair them, and take one from each pair. For other numbers, the long division method from NCERT Class 8 gives an exact answer, while estimation offers a quick approximation.<\/p>\n<p><strong>How do I find the square root of a non-perfect square?<\/strong><br \/>Use the long division method, which produces the root digit by digit and works for decimals too. Alternatively, estimate by finding the two nearest perfect squares and judging where your number falls between them.<\/p>\n<p><strong>Why do Indian exams ask for manual square root methods?<\/strong><br \/>Because calculators are banned in most board and competitive exams, students must compute roots by hand. The long division method is the standard NCERT technique tested in Class 8 and reused in later aptitude exams.<\/p>\n<p><strong>Does every number have two square roots?<\/strong><br \/>Every positive number has two real square roots, one positive and one negative, because a negative times a negative is positive. In school problems we usually take the positive, or principal, root unless the question says otherwise.<\/p>\n<p><script type=\"application\/ld+json\">{\"@context\":\"https:\/\/schema.org\",\"@type\":\"FAQPage\",\"mainEntity\":[{\"@type\":\"Question\",\"name\":\"What is the easiest way to find a square root?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"For perfect squares, prime factorisation is easiest: break the number into primes, pair them, and take one from each pair. For other numbers, the long division method gives an exact answer and estimation offers a quick approximation.\"}},{\"@type\":\"Question\",\"name\":\"How do I find the square root of a non-perfect square?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Use the long division method, which produces the root digit by digit and works for decimals. 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School problems usually take the positive principal root.\"}}]}<\/script><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Learn how to calculate square roots step by step: prime factorisation, NCERT long division and estimation, with examples.<\/p>\n","protected":false},"author":1,"featured_media":1534,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"footnotes":""},"categories":[24],"tags":[],"class_list":["post-1494","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>How to Calculate Square Root (Step by Step)<\/title>\n<meta name=\"description\" content=\"Learn how to calculate square roots step by step: prime factorisation, NCERT long division and estimation, with examples.\" \/>\n<meta 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