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The Cube Root of a number is a special value that, when multiplied by itself three times, gives you the original number. In simple terms, if you take a number and multiply it by itself two more times, the result is known as the cube of that number. This cube is represented as x3, which is read as “x cubed” or “x raised to the power of 3.”
For example, let’s consider the number 5. When you multiply 5 by itself three times (5 × 5 × 5), you get 125. Here, 125 is referred to as the cube of 5 and 5 is the cube root of 125.
Read this article to understand Cube Root in a better way.
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What is Cube Root?
The cube root of a number is the value that gives you the original number when cubbed. It’s represented by the symbol 3. Using our earlier example, the cube root of 125 is 5, because
5 × 5 × 5 = 125. it is also written as ∛125 = 5.
If we denote the cube root of a number x as y, the relationship can be expressed with the formula: ∛x = y
Here, the symbol ∛ represents the cube root, with a small 3 placed in the top left corner to indicate that it is a cube root specifically. Another way to represent the cube root is by using an exponent of 1/3. For example, x13 also denotes the cube root of x.
How to Use Cube Root?
The cube root of a number is represented by the symbol ∛. This symbol is used to denote the value that, when multiplied by itself three times, results in the original number. For example, to express the cube root of 27, we write:
27 = 3 3 3
This means that 3 is the number which, when multiplied by itself three times (3 × 3 × 3), gives us 27.
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What Are Perfect Cubes?
A perfect cube is an integer that can be written as the product of three identical integers. In simpler words, a number is a perfect cube if its cube root is an integer. For an example, 125 is a perfect cube because:
53 = 5 5 5 = 125
On the other hand, 121 is not a perfect cube, as there is no integer that can be multiplied by itself three times to result in 121.
| Number | Cube | Number | Cube |
| 1 | 1 | 6 | 216 |
| 2 | 8 | 7 | 343 |
| 3 | 27 | 8 | 512 |
| 4 | 64 | 9 | 729 |
| 5 | 125 | 10 | 1000 |
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How to Find the Cube Root of a Number?
Finding the cube root of a number is an easy process when using the prime factorization method. Follow the below-given steps to find the cube root.
Step 1: Begin with the prime factorization of the given number. Write the prime factors of the number.
Step 2: Group the prime factors into sets of three identical factors.
Step 3: Once grouped, remove the cube root symbol and multiply the factors within each group to get the answer. If a factor remains that cannot be grouped into threes, the original number is not a perfect cube, and its cube root cannot be found using this method.
Example: Finding the Cube Root of 15,625
| 5 | 15625 |
| 5 | 3125 |
| 5 | 625 |
| 5 | 125 |
| 5 | 25 |
| 5 | 5 |
| 1 |
15625 = 56 = 356 = 52 = 25
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Cube Root Formula
The cube root formula allows us to calculate the cube root of any given number, typically expressed as:
∛x = 3 y y y = y
In this formula, y represents the cube root of the number x. This also implies that x is a perfect cube if y is an integer.
Cube of Negative Numbers
The process for finding the cube of a negative number is the same as for a positive number or a fraction. The key difference is that the cube of a negative number will always be negative. This is because multiplying three negative numbers together results in a negative product.
For Example: Finding the Cube of -7
To calculate the cube of -7:
(-7) (-7) (-7) = 49 (-7) = -343
Therefore, the cube of -7 is -343.
Also, Cube Root of -343 is -7.
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Solved Examples of Cube Root
- 1. Find the cube of 6.
Ans. To find the cube of 6, multiply 6 by itself three times:
6 × 6 × 6 = 36 × 6 = 216
Answer: The cube of 6 is 216.
- 2. Find the cube root of 512.
Ans. Since 512 is a perfect cube, we can express it as:
512=(2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)=2 × 2 × 2 = 8
Answer: The cube root of 512 is 8.
- 3. Find the cube of 32.
Ans. To find the cube of 32, multiply 32 by itself three times:
32 × 32 × 32 = 278
Answer: The cube of 32 is 278.
- 4. Find the cube of -6.
Ans. To find the cube of -6, multiply -6 by itself three times:
-6 × -6 × -6 = 36 × (- 6) = -216
Answer: The cube of -6 is -216.
Practice Questions of Cube Root
1. Find the cube of 12.
2. Find the cube root of 27000.
3. Find the cube of 65.
4. Find the cube of -11.
5. Find the cube of 15.
6. Find the cube root of 729.
7. Find the cube of 59.
8. Find the cube of -4.
Cube Root: FAQs
What is a cube root?
The cube root of a number is a value that, when multiplied by itself three times, equals the original number. For example, the cube root of 27 is 3 because 3×3×3=27. It is denoted by the symbol ∛.
How do you calculate the cube of a fraction?
To find the cube of a fraction, multiply the fraction by itself three times. For example, to find the cube of 2/3, we multiply it by itself thrice. (2/3) × (2/3) × (2/3) = 8/27. Therefore, the cube of the fraction 2/3 is 8/27.
What happens when you cube a negative number?
When you cube a negative number, the result is always negative. This is because multiplying three negative numbers together results in a negative product. For example, the cube of -4 is -64.
