{"id":3704,"date":"2026-11-13T06:00:00","date_gmt":"2026-11-13T00:30:00","guid":{"rendered":"https:\/\/digitoolkit.in\/blog\/?p=3704"},"modified":"2026-11-13T06:00:00","modified_gmt":"2026-11-13T00:30:00","slug":"how-to-calculate-cube-root-step-by-step","status":"publish","type":"post","link":"https:\/\/digitoolkit.in\/blog\/how-to-calculate-cube-root-step-by-step\/","title":{"rendered":"How to Calculate Cube 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 find a cube root, ask which number multiplied by itself three times gives your number. For perfect cubes, use prime factorisation and group the factors in threes. For other numbers, estimate between two perfect cubes and refine, or use a free online cube root calculator for an exact decimal.<\/p>\n<p><strong>Key takeaways:<\/strong><\/p>\n<ul>\n<li>A cube root undoes cubing, so the cube root of 27 is 3.<\/li>\n<li>Prime factorisation works best for perfect cubes like 1728 or 9261.<\/li>\n<li>Estimation between two known cubes gives a quick answer for any other number.<\/li>\n<li>The cube root of a negative number is negative, so -64 gives -4.<\/li>\n<li>Always check your answer by cubing it again.<\/li>\n<\/ul>\n<\/div>\n<p>Your Class 8 teacher probably made you memorise cubes up to 20, and then asked you to go backwards. That backward step is the cube root. It shows up in school exams, in SSC and bank paper questions, and in real jobs like sizing a water tank. This guide shows you how to calculate it by hand, step by step, using simple Indian examples.<\/p>\n<p>You will learn three hand methods and one quick digital method. By the end, you should be able to take any number and decide which method is fastest for it. Let us start with the basic idea.<\/p>\n<h2>What Does a Cube Root Mean?<\/h2>\n<p>A cube is what you get when you multiply a number by itself three times. So 4 cubed is 4 times 4 times 4, which is 64. The cube root goes the other way. The cube root of 64 is 4, because 4 is the number that gives 64 when cubed.<\/p>\n<p>The sign for cube root is a small radical with a tiny 3 on it. Many textbooks, including the NCERT Class 8 chapter on cubes and cube roots, treat this as the opposite operation of cubing. You can think of it as asking, &#8220;What side length gives me this volume?&#8221; A cube-shaped box with volume 125 cubic centimetres has a side of 5 cm, because 5 times 5 times 5 is 125.<\/p>\n<p>Numbers like 8, 27, 64, 125 and 1000 are called perfect cubes. Their cube roots are whole numbers. Most other numbers, like 10 or 50, are not perfect cubes, so their cube roots are decimals that never end.<\/p>\n<h2>Method 1: Prime Factorisation (Best for Perfect Cubes)<\/h2>\n<p>This is the method your NCERT book teaches first, and it is the most reliable one for exams. You break the number into prime factors, then group equal factors in threes. Each group of three contributes one factor to the cube root. If any factor is left over in a group of one or two, the number is not a perfect cube.<\/p>\n<p>Follow these steps every time.<\/p>\n<ol>\n<li>Divide the number by the smallest prime (2, then 3, then 5, and so on) until you reach 1.<\/li>\n<li>Write all the prime factors in a row.<\/li>\n<li>Circle each group of three identical factors.<\/li>\n<li>Take one factor from each group.<\/li>\n<li>Multiply those factors to get the cube root.<\/li>\n<\/ol>\n<h3>Worked Example 1: A Cubic Water Tank in Litres<\/h3>\n<p>A housing society in Pune installs a cube-shaped water tank that holds 13,824 litres. We know 1000 litres equals 1 cubic metre, so the volume is 13.824 cubic metres. What is the length of one side?<\/p>\n<p>First find the cube root of 13824. Divide by 2 repeatedly: 13824 gives 6912, then 3456, then 1728, then 864, then 432, then 216, then 108, then 54, then 27. That is nine twos. Then 27 is 3 times 3 times 3. So 13824 equals nine twos and three threes.<\/p>\n<p>Group them in threes: three groups of 2 times 2 times 2, and one group of 3 times 3 times 3. Take one factor from each group: 2, 2, 2 and 3. Multiply to get 2 times 2 times 2 times 3, which is 24. So the cube root of 13824 is 24, and the cube root of 13.824 is 2.4.<\/p>\n<p>The tank side is 2.4 metres. Check: 2.4 times 2.4 is 5.76, and 5.76 times 2.4 is 13.824. The answer is correct.<\/p>\n<h2>Method 2: Estimation by Bracketing (For Any Number)<\/h2>\n<p>Prime factorisation fails when a number is not a perfect cube. Then you need an estimate. The idea is to find two perfect cubes that sit just below and just above your number, then guess where it falls between them.<\/p>\n<p>Suppose you want the cube root of 300. You know 6 cubed is 216 and 7 cubed is 343. Since 300 is between them, the answer lies between 6 and 7. It is closer to 343 than to 216, so try 6.7. Cubing 6.7 gives about 300.76, which is very close. So the cube root of 300 is about 6.69.<\/p>\n<p>To improve the guess, check the cube of your estimate. If it is too high, lower the estimate a little. If it is too low, raise it. Two or three rounds of this give two decimal places, which is enough for most school and exam work.<\/p>\n<h3>Worked Example 2: A Cube of Ice at a Kulfi Stall<\/h3>\n<p>A kulfi seller in Jaipur buys a block of ice that is shaped like a cube. Its volume is 9261 cubic centimetres. How long is each side?<\/p>\n<p>Bracket first. 20 cubed is 8000 and 22 cubed is 10648. So the side is between 20 and 22. The last digit of 9261 is 1, and a number ending in 1 can only come from a cube root ending in 1. So try 21. Now 21 times 21 is 441, and 441 times 21 is 9261. The side is exactly 21 cm.<\/p>\n<p>Notice how the last digit narrowed the options. That small trick is the heart of the Vedic shortcut that many SSC aspirants use, and we have covered it separately in our beginner examples guide.<\/p>\n<h2>Method 3: Using a Calculator for Instant Answers<\/h2>\n<p>When the number is large or has decimals, hand methods become slow. A <a href=\"https:\/\/digitoolkit.in\/calculators\/cube-root-calculator\/\">cube root calculator<\/a> gives the answer in a second, usually to many decimal places. You type the number, press calculate, and read the result. This is useful for checking homework, planning a project, or confirming your own working.<\/p>\n<p>Use the tool as a checker, not as a replacement for understanding. Do the hand method first, then compare. If both match, you have learned the method. If they differ, you have found a mistake worth fixing.<\/p>\n<h3>Worked Example 3: A Gold Cube and Its Price in INR<\/h3>\n<p>A jeweller wants a solid gold cube with a volume of 125 cubic centimetres for a display piece. First, the side. The cube root of 125 is 5, because 5 times 5 times 5 is 125. So each side is 5 cm.<\/p>\n<p>Now the weight. Pure gold has a density of about 19.32 grams per cubic centimetre. So the cube weighs 125 times 19.32, which is 2415 grams. Assume, only for this example, a price of Rs. 7,000 per gram. Then the value is 2415 times 7000, which is Rs. 1,69,05,000. Real prices change daily, so always use the current rate and remember that jewellery also carries making charges and GST.<\/p>\n<h2>Quick Reference Table of Perfect Cubes<\/h2>\n<p>Memorising these pairs makes every method faster. Learn them once and you will use them for years.<\/p>\n<table>\n<thead>\n<tr>\n<th>Number<\/th>\n<th>Cube<\/th>\n<th>Cube root of the cube<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>2<\/td>\n<td>8<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>27<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>64<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>125<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>216<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>343<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>512<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>729<\/td>\n<td>9<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>1000<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>1728<\/td>\n<td>12<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<blockquote>\n<p><strong>Key takeaway:<\/strong> Use prime factorisation when you suspect a perfect cube, estimation when you do not, and a calculator to verify. Choosing the right method saves more time than calculating faster.<\/p>\n<\/blockquote>\n<h2>Cube Roots of Negative Numbers and Decimals<\/h2>\n<p>Unlike square roots, cube roots work fine for negative numbers. The cube root of -27 is -3, because -3 times -3 times -3 equals -27. Two negatives make a positive, and a third negative makes it negative again. So the sign of the answer always matches the sign of the number.<\/p>\n<p>For decimals, count the decimal places. A number with three decimal places, like 0.008, has a cube root with one decimal place, which is 0.2. A number with six decimal places, like 0.000064, has a cube root with two, which is 0.04. Group the digits in threes from the decimal point to keep track.<\/p>\n<h2>Benefits of Learning Cube Root Methods<\/h2>\n<p>Knowing how to find a cube root by hand builds number sense. You start to see patterns in cubes, and that helps in algebra, mensuration and data interpretation. Students who practise these methods usually find volume problems less scary.<\/p>\n<ul>\n<li>Faster solving in SSC, banking and NTSE style questions.<\/li>\n<li>Better understanding of volume, since side and volume are linked by cubing.<\/li>\n<li>Fewer errors, because you can check any answer by cubing it.<\/li>\n<li>Confidence in exams where calculators are not allowed.<\/li>\n<li>Practical use in planning tanks, boxes, and storage in daily life.<\/li>\n<\/ul>\n<h2>Challenges and Limitations<\/h2>\n<p>Each method has limits. Prime factorisation is slow for very large numbers, and it only gives a clean answer for perfect cubes. Estimation takes patience, and your accuracy depends on how well you know your cubes. Even a calculator can mislead you if you type a wrong digit.<\/p>\n<p>Another challenge is that most cube roots are irrational. The cube root of 2 is about 1.26, but the digits never end or repeat. So any answer you write for such numbers is an approximation, and you should say how many decimal places you used.<\/p>\n<h2>Common Mistakes to Avoid<\/h2>\n<ul>\n<li><strong>Confusing cube root with square root.<\/strong> The square root of 64 is 8, but the cube root of 64 is 4, so read the question carefully.<\/li>\n<li><strong>Dividing by 3.<\/strong> The cube root of 27 is 3, not 9, because you are not dividing the number by 3.<\/li>\n<li><strong>Grouping factors in twos.<\/strong> Pairs belong to square roots, and cube roots need groups of three.<\/li>\n<li><strong>Dropping the negative sign.<\/strong> The cube root of -8 is -2, not 2.<\/li>\n<li><strong>Skipping the check.<\/strong> Always cube your answer once, because it takes ten seconds and catches most errors.<\/li>\n<li><strong>Mixing up units.<\/strong> A volume in litres must be changed to cubic metres or cubic centimetres first, remembering that 1 litre is 1000 cubic centimetres.<\/li>\n<\/ul>\n<h2>Best Practices for Accurate Results<\/h2>\n<ol>\n<li>Memorise cubes from 1 to 20 so brackets come to mind instantly.<\/li>\n<li>Look at the last digit of the number first, since it tells you the last digit of the cube root.<\/li>\n<li>Write prime factors in a neat column so you do not miss any.<\/li>\n<li>Use a <a href=\"https:\/\/digitoolkit.in\/calculators\/square-root-calculator\/\">square root calculator<\/a> to compare square and cube behaviour when you are unsure which one a question wants.<\/li>\n<li>For tank and box problems, convert units before you calculate, and a volume calculator helps you confirm the final volume.<\/li>\n<li>State your answer with units and the right number of decimal places.<\/li>\n<\/ol>\n<h2>Conclusion<\/h2>\n<p>Calculating a cube root is a skill with a clear process. Use prime factorisation for perfect cubes, bracket and refine for other numbers, and verify with a calculator. Practise with real objects like tanks, ice blocks and gold cubes, and the idea will stick.<\/p>\n<p>Start today with five numbers from the table above. Find each cube root by hand, then check it online. Within a week, you will be faster than you expect.<\/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\/cube-root-calculator\/\">Try the free Cube Root Calculator &rarr;<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/cube-root-formula-explained-with-examples\/\">Cube Root Formula Explained with Examples<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/what-is-a-cube-root-simple-guide\/\">What Is a Cube Root? A Simple Guide<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/cube-root-calculator-free-online-tool-guide\/\">Cube Root Calculator: Free Online Tool and Guide<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/cube-root-examples-for-beginners\/\">Cube 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\/exponent-calculator-free-online-tool-guide\/\">Exponent Calculator: Free Online Tool + Complete Guide<\/a><\/li>\n<li><a href=\"https:\/\/digitoolkit.in\/blog\/category\/mathematics\/\">More Mathematics guides<\/a><\/li>\n<\/ul>\n<\/div>\n<h2>FAQs<\/h2>\n<p><strong>What is the easiest way to find a cube root by hand?<\/strong><\/p>\n<p>For perfect cubes, prime factorisation is the most dependable. Break the number into primes, group them in threes, and multiply one factor from each group. For other numbers, bracket the answer between two known cubes and refine it.<\/p>\n<p><strong>Can a cube root be negative?<\/strong><\/p>\n<p>Yes. The cube root of a negative number is negative, because three negative factors multiply to a negative result. For example, the cube root of -125 is -5.<\/p>\n<p><strong>How do I know if a number is a perfect cube?<\/strong><\/p>\n<p>Factorise it into primes. If every prime appears a multiple of three times, it is a perfect cube. For example, 1728 has six twos and three threes, so it is a perfect cube.<\/p>\n<p><strong>Is the cube root of 1000 equal to 10?<\/strong><\/p>\n<p>Yes. Ten times ten times ten is 1000, so the cube root of 1000 is exactly 10. This is why 1000 litres make a cube of side one metre.<\/p>\n<p><strong>Can I use a calculator in exams like SSC or banking?<\/strong><\/p>\n<p>Most competitive exams do not allow personal calculators, so hand methods matter. Use an online tool for practice and for checking your work at home.<\/p>\n<p><strong>What is the cube root of a decimal like 0.027?<\/strong><\/p>\n<p>Treat it as 27 divided by 1000. The cube root of 27 is 3 and of 1000 is 10, so the answer is 0.3.<\/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 cube root by hand?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"For perfect cubes, prime factorisation is the most dependable. Break the number into primes, group them in threes, and multiply one factor from each group. For other numbers, bracket the answer between two known cubes and refine it.\"}},{\"@type\":\"Question\",\"name\":\"Can a cube root be negative?\",\"acceptedAnswer\":{\"@type\":\"Answer\",\"text\":\"Yes. The cube root of a negative number is negative, because three negative factors multiply to a negative result. 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