What are Perfect Cubes?
When an integer is multiplied by itself three times, the integer that we get is called a perfect cube. A perfect cube is the value of the power when the exponent is 3. Let us understand this with an example: when we multiply 2 three times, we get 8, and 8 is said to be a perfect cube.
This is represented mathematically as \( 2\times 2\times 2=2^3=8 \).
Similarly, \( 3\times 3\times 3=3^3=27 \), so 27 is also a perfect cube.
Perfect Cubes of 1 to 20
Perfect Cubes of 21 to 40
Perfect Cubes of 41 to 50
Signs of Perfect Cubes
A perfect cube can be positive or negative. Let us find the cubes of some positive and negative numbers.
(- 5) \( \times \) (- 5) \( \times \) (- 5) = – 125
5 \( \times \) 5 \( \times \) 5 = 125
2 \( \times \) 2 \( \times \) 2 = 8
(- 2) \( \times \) (- 2) \( \times \) (- 2) = – 8
Therefore, we can say that-
- The cube of a negative number is always negative.
- The cube of a positive number is always positive.
How can we check for a Perfect Cube?
To check whether a number is a perfect cube or not, we use the prime factorization method. Let us check whether 216 is a perfect cube or not.
Step 1: The factor tree of 216 is as below:
The prime factorization of 216 = 2 \( \times \) 2 \( \times \) 2 \( \times \) 3 \( \times \) 3 \( \times \) 3
Step 2: Group the same factors into groups of 3
216 = (2 \( \times \) 2 \( \times \) 2) \( \times \) (3 \( \times \) 3 \( \times \) 3)
Step 3: Check if any factor is left ungrouped.
Here, no factor is left ungrouped, so, 216 is a perfect cube.
Solved Examples
Example 1: Is 512 a perfect cube?
Solution: Let us create the factor tree of 512:
The prime factorization of 512 = 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2
The prime factorization of 512 = (2 \( \times \) 2 \( \times \) 2) \( \times \) (2 \( \times \) 2 \( \times \) 2) \( \times \) (2 \( \times \) 2 \( \times \) 2)
Here, all the factors are grouped together, so 512 is a perfect cube.
Example 2: Is 256 a perfect cube? If not, what should be multiplied to make it a perfect cube?
Solution: Let us create the factor tree of 256:
The prime factorization of 256 = 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2 \( \times \) 2
The prime factorization of 256 = (2 \( \times \) 2 \( \times \) 2) \( \times \) (2 \( \times \) 2 \( \times \) 2) \( \times \) 2 \( \times \) 2
Here, all the 2s are not in groups of three, so 256 is not a perfect cube.
To group all factors into groups of 3, we would need one more 2.
Therefore, 2 should be multiplied by 256 to make it a perfect cube.
Yes, a cube of any negative number is always a negative number.
Perfect cubes are both even and odd. The perfect cube of an even number is always even, and the perfect cube of an odd number is always odd.
Let us find the cubes of the first 6 numbers:
1\( ^3 \) = 1, 2\( ^3 \) = 8, 3\( ^3 \) = 27, 4\( ^3 \) = 64, 5\( ^3 \) = 125, 6\( ^3 \) = 216
From above, we can see that only one perfect cube lies between 100 and 200, which is 125.