Finding Cube Roots
Cube roots are a natural extension of the concept of square roots, introduced in Class 8 under the chapter Cubes and Cube Roots. While a square root asks 'what number multiplied by itself gives n?', a cube root asks 'what number multiplied by itself three times gives n?' For example, the cube root of 27 is 3 because 3 x 3 x 3 = 27, and the cube root of 125 is 5 because 5 x 5 x 5 = 125. Cube roots play an important role in mathematics and science — they appear in volume calculations (finding the side of a cube given its volume), in scientific formulas, and in understanding three-dimensional geometry. Unlike square roots, cube roots can be found for both positive and negative numbers. The cube root of -8 is -2 because (-2) x (-2) x (-2) = -8. This makes cube roots fundamentally different from square roots in their behaviour with negative numbers. In Class 8, students learn to find cube roots primarily through the prime factorisation method and through estimation techniques. Understanding cube roots also helps in recognising patterns in numbers, simplifying algebraic expressions, and solving equations involving cubes. The concept naturally leads into the study of exponents, logarithms, and higher-order roots in subsequent classes.
What is Finding Cube Roots?
The cube root of a number is a value that, when multiplied by itself three times (cubed), gives the original number. If x x x x x = n (that is, x cubed = n), then x is the cube root of n.
The cube root of n is denoted by the symbol n^(1/3) or written with the cube root radical as the third root of n.
Key concepts:
Perfect cubes: Numbers that are cubes of whole numbers are called perfect cubes. The first few perfect cubes are:
1 cubed = 1, 2 cubed = 8, 3 cubed = 27, 4 cubed = 64, 5 cubed = 125, 6 cubed = 216, 7 cubed = 343, 8 cubed = 512, 9 cubed = 729, 10 cubed = 1000.
Non-perfect cubes: Numbers like 2, 3, 4, 5, 6, 7, 9, 10, 11, etc. (those not in the list above) are not perfect cubes. Their cube roots are irrational numbers. For example, the cube root of 2 is approximately 1.2599.
Cube roots of negative numbers: Unlike square roots, cube roots of negative numbers are defined in the real number system. The cube root of a negative number is negative. For example:
The cube root of -27 = -3, because (-3) x (-3) x (-3) = -27.
The cube root of -64 = -4, because (-4) x (-4) x (-4) = -64.
Cube root of zero: The cube root of 0 is 0, because 0 x 0 x 0 = 0.
Uniqueness: Every real number has exactly one real cube root. This is different from square roots, where positive numbers have two roots (positive and negative). The cube root of 8 is only 2 (not -2, since (-2) cubed = -8, not 8).
Methods
There are three main methods to find cube roots at the Class 8 level:
Method 1: Prime Factorisation Method
This is the most commonly used method for finding cube roots of perfect cubes. Steps:
1. Find the complete prime factorisation of the given number.
2. Group the prime factors into triplets (groups of three identical factors).
3. Take one factor from each triplet.
4. Multiply these factors together to get the cube root.
If any prime factor does not form a complete triplet, the number is NOT a perfect cube.
Example: Cube root of 1728.
1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3
Triplets: (2, 2, 2), (2, 2, 2), (3, 3, 3)
One from each: 2, 2, 3
Cube root = 2 x 2 x 3 = 12.
Method 2: Estimation Method
For large perfect cubes or approximate cube roots, use estimation. Steps:
1. Identify the last digit of the number and use the units digit pattern of cubes to determine the units digit of the cube root.
2. Remove the last three digits and find the largest perfect cube less than or equal to the remaining number. This gives the tens digit.
3. Combine to get the cube root.
Units digit pattern for cubes:
If a cube ends in 0, cube root ends in 0.
If a cube ends in 1, cube root ends in 1.
If a cube ends in 2, cube root ends in 8.
If a cube ends in 3, cube root ends in 7.
If a cube ends in 4, cube root ends in 4.
If a cube ends in 5, cube root ends in 5.
If a cube ends in 6, cube root ends in 6.
If a cube ends in 7, cube root ends in 3.
If a cube ends in 8, cube root ends in 2.
If a cube ends in 9, cube root ends in 9.
Notice that 0, 1, 4, 5, 6, 9 repeat themselves, while 2 and 8 swap, and 3 and 7 swap.
Method 3: Successive Estimation for Non-Perfect Cubes
For non-perfect cubes, find the two consecutive integers whose cubes bracket the given number. Then refine using trial and improvement or interpolation.
Example: Cube root of 100. Since 4 cubed = 64 and 5 cubed = 125, the cube root of 100 lies between 4 and 5. By further trial, 4.6 cubed = 97.336 and 4.7 cubed = 103.823, so the cube root of 100 is approximately 4.64.
Solved Examples
Example 1: Example 1: Cube root by prime factorisation
Problem: Find the cube root of 5832.
Solution:
Step 1: Find the prime factorisation of 5832.
5832 / 2 = 2916
2916 / 2 = 1458
1458 / 2 = 729
729 / 3 = 243
243 / 3 = 81
81 / 3 = 27
27 / 3 = 9
9 / 3 = 3
3 / 3 = 1
5832 = 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3
Step 2: Group into triplets: (2, 2, 2), (3, 3, 3), (3, 3, 3).
Step 3: Take one from each triplet: 2, 3, 3.
Step 4: Multiply: 2 x 3 x 3 = 18.
Answer: The cube root of 5832 is 18.
Example 2: Example 2: Cube root of a negative number
Problem: Find the cube root of -2744.
Solution:
Step 1: Since the number is negative, the cube root will also be negative.
The cube root of -2744 = -(the cube root of 2744).
Step 2: Find the prime factorisation of 2744.
2744 / 2 = 1372
1372 / 2 = 686
686 / 2 = 343
343 / 7 = 49
49 / 7 = 7
7 / 7 = 1
2744 = 2 x 2 x 2 x 7 x 7 x 7
Step 3: Triplets: (2, 2, 2), (7, 7, 7). One from each: 2, 7.
Step 4: Cube root of 2744 = 2 x 7 = 14.
Step 5: The cube root of -2744 = -14.
Answer: The cube root of -2744 is -14.
Example 3: Example 3: Cube root by estimation method
Problem: Find the cube root of 19683 using the estimation method.
Solution:
Step 1: Look at the units digit. 19683 ends in 3. From the units digit pattern, the cube root ends in 7.
Step 2: Remove the last three digits. We are left with 19.
Step 3: Find the largest perfect cube less than or equal to 19.
2 cubed = 8, 3 cubed = 27. Since 8 is less than or equal to 19 but 27 is greater than 19, the tens digit is 2.
Step 4: Combine: The cube root is 27.
Step 5: Verify: 27 x 27 x 27 = 27 x 729 = 19683. Correct!
Answer: The cube root of 19683 is 27.
Example 4: Example 4: Checking if a number is a perfect cube
Problem: Is 3600 a perfect cube? If not, find the smallest number by which it must be multiplied to make it a perfect cube.
Solution:
Step 1: Find the prime factorisation of 3600.
3600 / 2 = 1800
1800 / 2 = 900
900 / 2 = 450
450 / 2 = 225
225 / 3 = 75
75 / 3 = 25
25 / 5 = 5
5 / 5 = 1
3600 = 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5
Step 2: Group into triplets:
2: appears 4 times. One triplet (2,2,2) and one extra 2.
3: appears 2 times. Needs one more 3.
5: appears 2 times. Needs one more 5.
Step 3: 3600 is NOT a perfect cube because not all factors form complete triplets.
Step 4: To make it a perfect cube, multiply by: 2 squared x 3 x 5 = 4 x 3 x 5 = 60.
Wait, we need 2 more 2's? Let me recheck. 2 appears 4 times: one triplet uses 3, leaving 1 extra. So we need 2 more 2's. 3 appears 2 times: need 1 more. 5 appears 2 times: need 1 more.
Multiply by: 2 x 2 x 3 x 5 = 60.
Step 5: 3600 x 60 = 216000. Check: 216000 = 60 cubed = 60 x 60 x 60 = 216000. The cube root is 60.
Answer: 3600 is not a perfect cube. Multiply by 60 to get 216000, whose cube root is 60.
Example 5: Example 5: Cube root of a fraction
Problem: Find the cube root of 729/1000.
Solution:
Step 1: Use the quotient property.
The cube root of (729/1000) = the cube root of 729 / the cube root of 1000.
Step 2: Find each cube root.
729 = 9 x 9 x 9, so the cube root of 729 = 9.
1000 = 10 x 10 x 10, so the cube root of 1000 = 10.
Step 3: The cube root of (729/1000) = 9/10 = 0.9.
Answer: The cube root of 729/1000 is 9/10 or 0.9.
Example 6: Example 6: Finding the side of a cube given its volume
Problem: The volume of a cube is 13824 cubic cm. Find the side length of the cube.
Solution:
Step 1: Volume of a cube = side cubed. So, side = the cube root of volume.
Step 2: Find the prime factorisation of 13824.
13824 / 2 = 6912
6912 / 2 = 3456
3456 / 2 = 1728
1728 / 2 = 864
864 / 2 = 432
432 / 2 = 216
216 / 2 = 108
108 / 2 = 54
54 / 2 = 27
27 / 3 = 9
9 / 3 = 3
3 / 3 = 1
13824 = 2 to the power 9 x 3 to the power 3
Step 3: Triplets: (2,2,2), (2,2,2), (2,2,2), (3,3,3). All paired.
Step 4: One from each triplet: 2, 2, 2, 3.
Side = 2 x 2 x 2 x 3 = 24 cm.
Answer: The side of the cube is 24 cm.
Example 7: Example 7: Product property of cube roots
Problem: Find the cube root of 8000 using the product property.
Solution:
Step 1: Express 8000 as a product of numbers whose cube roots are known.
8000 = 8 x 1000.
Step 2: Apply the product property.
The cube root of 8000 = the cube root of 8 x the cube root of 1000
= 2 x 10 = 20.
Step 3: Verify: 20 x 20 x 20 = 8000. Correct!
Answer: The cube root of 8000 is 20.
Example 8: Example 8: Cube root of a decimal
Problem: Find the cube root of 0.000216.
Solution:
Step 1: Express as a fraction.
0.000216 = 216 / 1000000
Step 2: Find cube roots of numerator and denominator.
216 = 6 x 6 x 6, so the cube root of 216 = 6.
1000000 = 100 x 100 x 100, so the cube root of 1000000 = 100.
Step 3: The cube root of 0.000216 = 6 / 100 = 0.06.
Answer: The cube root of 0.000216 is 0.06.
Example 9: Example 9: Finding the smallest divisor to make a perfect cube
Problem: Find the smallest number by which 1536 must be divided so that the quotient is a perfect cube.
Solution:
Step 1: Find the prime factorisation of 1536.
1536 / 2 = 768
768 / 2 = 384
384 / 2 = 192
192 / 2 = 96
96 / 2 = 48
48 / 2 = 24
24 / 2 = 12
12 / 2 = 6
6 / 2 = 3
3 / 3 = 1
1536 = 2 to the power 9 x 3
Step 2: Check triplets:
2 appears 9 times = 3 complete triplets. Perfect.
3 appears 1 time. It needs 2 more to complete a triplet, or we can divide by 3 to remove it.
Step 3: Divide 1536 by 3 to get 512.
512 = 2 to the power 9 = (2 cubed) cubed = 8 cubed.
Answer: The smallest number is 3. Dividing 1536 by 3 gives 512, whose cube root is 8.
Example 10: Example 10: Comparing cube roots
Problem: Without calculating the actual values, determine which is greater: the cube root of 729 or the cube root of 512.
Solution:
Step 1: Recall that 729 = 9 cubed and 512 = 8 cubed.
Step 2: The cube root of 729 = 9 and the cube root of 512 = 8.
Step 3: Since 9 is greater than 8, the cube root of 729 is greater.
Alternatively, since 729 is greater than 512 and the cube root function is an increasing function (larger input gives larger output), the cube root of 729 must be greater.
Answer: The cube root of 729 (= 9) is greater than the cube root of 512 (= 8).
Real-World Applications
Cube roots have many practical applications in mathematics, science, and everyday life:
Volume and Geometry: The most direct application is finding the side of a cube when its volume is known. If a cube has volume V, then each side = the cube root of V. This extends to other 3D shapes where cube root relationships appear in scaling.
Scaling and Proportions: When objects are scaled up or down in all three dimensions, volumes change as the cube of the scaling factor. To find the scaling factor from a volume ratio, you take the cube root. For example, if one container holds 8 times the volume of another, it is the cube root of 8 = 2 times as large in each dimension.
Science and Chemistry: In chemistry, the relationship between the volume of a gas and the number of molecules involves cube roots. The concept of molar volume and molecular spacing uses cube root calculations.
Engineering: Engineers use cube roots when designing containers, tanks, and storage units. If a tank needs to hold a certain volume, the cube root helps determine the minimum dimensions needed.
Physics: The intensity of gravitational and electromagnetic fields follows inverse-square and inverse-cube laws. Cube roots appear when solving for distances in these contexts.
Statistics and Data Analysis: Cube roots are used as a data transformation technique to normalise skewed data distributions, making statistical analysis more reliable.
Key Points to Remember
- The cube root of a number n is a value that, when cubed (multiplied by itself three times), gives n.
- Every real number has exactly one real cube root — positive numbers have positive cube roots, negative numbers have negative cube roots, and the cube root of 0 is 0.
- Unlike square roots, cube roots of negative numbers are defined in the real number system.
- The prime factorisation method finds cube roots by grouping prime factors into triplets of three identical factors.
- If a prime factor cannot be grouped into a complete triplet, the number is not a perfect cube.
- The estimation method uses the units digit pattern: 0-0, 1-1, 2-8, 3-7, 4-4, 5-5, 6-6, 7-3, 8-2, 9-9.
- The cube root of (a x b) = the cube root of a multiplied by the cube root of b (product property).
- The cube root of (a / b) = the cube root of a divided by the cube root of b (quotient property).
- The cube root of (-n) = -(the cube root of n) — the negative sign can be taken outside.
- Perfect cubes up to 10: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
Practice Problems
- Find the cube root of 10648 using the prime factorisation method.
- Find the cube root of -32768 using prime factorisation.
- Is 5400 a perfect cube? If not, find the smallest number by which it must be multiplied to get a perfect cube.
- Find the cube root of 27000 using the product property.
- The volume of a cubical tank is 74088 litres. Find the side of the tank in decimetres.
- Find the cube root of 0.001728 by expressing it as a fraction.
- A number is 5 times the cube root of 1728. Find the number.
- Find the cube root of 46656 using the estimation method.
Frequently Asked Questions
Q1. What is a cube root?
A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 64 is 4 because 4 x 4 x 4 = 64. It is the reverse operation of cubing a number.
Q2. What is the difference between a square root and a cube root?
A square root of n is a number that, when multiplied by itself (twice), gives n. A cube root of n is a number that, when multiplied by itself three times, gives n. For example, the square root of 16 is 4 (since 4 x 4 = 16), while the cube root of 27 is 3 (since 3 x 3 x 3 = 27). Also, cube roots exist for negative numbers (cube root of -8 = -2), while real square roots of negative numbers do not exist.
Q3. Can we find the cube root of a negative number?
Yes. Unlike square roots, cube roots of negative numbers are real numbers. The cube root of a negative number is always negative. For example, the cube root of -125 = -5 because (-5) x (-5) x (-5) = -125. This is possible because the product of three negative numbers is negative.
Q4. How do you find the cube root by prime factorisation?
First, find the complete prime factorisation of the number. Then group the prime factors into triplets (groups of three identical factors). Take one factor from each triplet and multiply them together. The result is the cube root. If any factor does not form a complete triplet, the number is not a perfect cube.
Q5. What are perfect cubes?
Perfect cubes are numbers that result from multiplying a whole number by itself three times. The first ten perfect cubes are: 1 (1 cubed), 8 (2 cubed), 27 (3 cubed), 64 (4 cubed), 125 (5 cubed), 216 (6 cubed), 343 (7 cubed), 512 (8 cubed), 729 (9 cubed), and 1000 (10 cubed).
Q6. How can you tell the units digit of a cube root from the number?
There is a pattern: if a perfect cube ends in 0, its cube root ends in 0; ends in 1, root ends in 1; ends in 2, root ends in 8; ends in 3, root ends in 7; ends in 4, root ends in 4; ends in 5, root ends in 5; ends in 6, root ends in 6; ends in 7, root ends in 3; ends in 8, root ends in 2; ends in 9, root ends in 9.
Q7. Is the cube root of 2 a rational number?
No, the cube root of 2 is an irrational number. Its value is approximately 1.2599 and the decimal goes on forever without repeating. In general, the cube root of any non-perfect cube is irrational.
Q8. What is the cube root of 1?
The cube root of 1 is 1, because 1 x 1 x 1 = 1. In the real number system, 1 is the only cube root of 1.
Q9. How is finding cube roots useful in real life?
Cube roots are used to find the side length of a cube when its volume is known, to understand scaling in three dimensions (e.g., if you double the volume, each dimension increases by the cube root of 2), and in scientific calculations involving density, molecular spacing, and gravitational fields.
Q10. Can the cube root of a number be a fraction?
Yes. For example, the cube root of 1/8 is 1/2 because (1/2) x (1/2) x (1/2) = 1/8. The cube root of a fraction equals the cube root of the numerator divided by the cube root of the denominator.
