Generally, a cube is defined as a solid figure that has all sides equal. In other words, a Cube is a solid three-dimensional figure that consists of 6 square faces, 8 vertices, and 12 edges. When six identical square face together along with their edges form a cube. Also, the three edges of a cube join at each corner to form a vertex. The cube in a Cube and Cube Roots chapter is the first and basic concept that will let you know about the perfect cubes or cube numbers.
In mathematical terms, a cube of a number is explained as the multiplication of a number by three times. The cube of a number represents as a superscript 3 or 3 is written a little up to the right of the numbers.
Representation of a cube of a number: m³, where 3 is the power of m and read as “m cubed”.
Cube of a Number
A number is multiplied by itself 3 times to find the cube of that number.
cube of m = m × m × m
cube of m = m³
Examples:
(i) Find the cube of a number 3?
Solution:
cube of 3 = 3³
The cube of 3 = 3³ = 3 × 3 × 3
cube of 3 = 3³ = 27
The cube of 3 is 27
(ii) Find the cube of a number 2?
Solution:
Cube of 2 = 2³
The cube of 2 = 2³ = 2 × 2 × 2
cube of 2 = 2³ = 8
The cube of 2 is 8
(iii) Find the cube of a number 4?
Solution:
cube of 4 = 4³
The cube of 4 = 4³ = 4 × 4 × 4
cube of 4 = 4³ = 64
The cube of 4 is 64
(iii) Find the cube of a number 5?
Solution:
Cube of 5 = 5³
The cube of 5 = 5³ = 5 × 5 × 5
cube of 5 = 5³ = 125
The cube of 5 is 125
(iv) Find the cube of a number 6?
Solution:
cube of 6 = 6³
The cube of 6 = 6³ = 6 × 6 × 6
cube of 6 = 6³ = 216
The cube of 6 is 216
(v) Find the cube of a number 7?
Solution:
cube of 7 = 7³
The cube of 7 = 7³ = 7 × 7 × 7
cube of 7 = 7³ = 343
The cube of 7 is 343
Perfect Cubes and Cube Roots
A perfect cube is defined as a number that is the cube of an integer or the cube of some natural number.
Examples:
Cube of 1 = 1³ = 1 × 1 × 1 = 1
Cube of 2 = 2³ = 2 × 2 × 2 = 8
The Cube of 3 = 3³ = 3 × 3 × 3 = 27
Cube of 4 = 4³ = 4 × 4 × 4 = 64
The Cube of 5 = 5³ = 5 × 5 × 5 = 125
Cube of Negative Numbers
Cubing the number is nothing but raising the number to its 3rd power. The Cube of a negative number is always a negative number. If -m is a number, then the cube of -m is (-m)³ = -m × -m × -m
Examples:
(i) Find the cube of -1?
Answer:
cube of -1 = (-1)³ = -1 × -1 × -1 = -1
The cube of -1 is -1
(ii) Find the cube of -2?
Answer:
cube of -2 = (-2)³ = -2 × -2 × -2 = -8
The cube of -2 is -8
(iii) Find the cube of -3?
Answer:
cube of -3 = (-3)³ = -3 × -3 × -3 = -27
The cube of -3 is -27
(iv) Find the cube of -4?
Answer:
cube of -4 = (-4)³ = -4 × -4 × -4 = -64
The cube of -4 is -64
(v) Find the cube of -5?
Answer:
cube of -5 = (-5)³ = -5 × -5 × -5 = -125
The cube of -5 is -125
Cube of a Rational Number
Finding the cube of a rational number is represents as the (a/b)³ that is also eqaul to the a/b × a/b × a/b = (a × a × a)/(b × b × b) = a³/b³
Therefore, (a/b)³ = a³/b³
Examples:
(i) Find the cube of (1/2)³?
Answer:
cube of 1/2 = (1/2)³
The cube of 1/2 = (1/2)³ = (1/2) × (1/2) × (1/2)
cube of 1/2 = 1³/2³
cube of 1/2 = (1/2)³ = 1³/2³ = (1 × 1 × 1)/(2 × 2 × 2)
The cube of 1/2 = (1/2)³ = 1³/2³ = 1/8
The cube of (1/2) is 1/8
(i) Find the cube of (-4/3)³?
Answer:
cube of (-4/3) = (-4/3)³
The cube of (-4/3) = (-4/3)³ = (-4/3) × (-4/3) × (-4/3)
cube of (-4/3) = (-4)³/3³
cube of (-4/3) = (-4/3)³ = (-4)³/3³ = (-4 × -4 × -4)/(3 × 3 × 3)
The cube of (-4/3) = (-4/3)³ = (-4)³/3³ = -64/27
The cube of (-4/3) is -64/27
Cube number Properties:
(i) The cube of even integers is always even.
(ii) The cube of odd integers is always odd.
Perfect Cube Solved examples
1. Is 189 a perfect cube?
Answer:
Separate 189 into different prime factors
The prime factors for 189 are 3, 3, 3, 7
189 = 3 × 3 × 3 × 7
189 = 3 × 7
Divide 189 with 7 to make it a perfect cube.
So, 189 is not a perfect cube.
2. Find the number 216 is a perfect cube?
Answer:
Firstly, find the prime factors of 216
The prime factors of 216 are 2, 2, 2, 3, 3, 3
So, 216 = 2 × 2 × 2 × 3 × 3 × 3
216 = (2 × 3) × (2 × 3) × (2 × 3)
216 = 6 × 6 × 6
The 216 = 6³ = cube of 6
216 is a perfect cube as it is the cube of 6.
216 is a perfect cube
3. Find the smallest number that makes the 3087 a perfect cube?
Answer:
To know the smallest number that makes the 3087 a perfect cube, first, we need to find the prime factors of 3087.
The 3087 prime factors are 3, 3, 7, 7, 7
3087 = 3 × 3 × 7 × 7 × 7
If the product of prime factors is multiplied by the number 3, then 3087 becomes a perfect cube.
The required number is 3
4. Which number needs to divide from 392 to make it a perfect cube?
Answer:
To find the number that makes 392 a perfect cube, we need to find the prime factors of 392
The prime factors of 392 are 2, 2, 2, 7, 7
392 = 2 × 2 × 2 × 7 × 7
The 392 will becomes a perfect cube if 7 × 7 is divided from the product of prime factors.
The required number is 7 × 7
5. Find the cube of each of the following?
(i) 8 (ii) (2/5) (iii) 0.2 (iv) 2 3/4 (v) -5
Solutions:
(i) 8
cube of 8 = 8³
The cube of 8 = 8³ = 8 × 8 × 8
cube of 8 = 8³ = 512
The cube of 8 is 512
(i)(2/5)
cube of (2/5) = (2/5)³
The cube of (2/5) = (2/5)³ = (2/5) × (2/5) × (2/5)
cube of (2/5) = (2/5)³ = 2³/5³ = (2 × 2 × 2)/(5 × 5 × 5) = 8/125
The cube of (2/5) is 8/125
(iii) 0.2
cube of 0.2 = 0.2³
The cube of 0.2 = 0.2³ = (0.2) × 0.2 × 0.2
cube of 0.2 = (0.2)³ = 0.008
The cube of (0.2) is 0.008
(iv) 2 3/4
2 3/4 = 11/4
cube of 11/4 = (11/4)³
The cube of (11/4) = (11/4)³ = (11/4) × (11/4) × (11/4)
cube of (11/4) = (11/4)³ = (11 × 11 × 11)/(4 × 4 × 4) = 1331/64
The cube of (2 3/4) is 1331/64
(i) -5
cube of -5 = -5³
The cube of -5 = -5³ = -5 × -5 × -5
cube of -5 = -5³ = -125
The cube of (-5) is -125
The cube of a number is clearly explained along with examples and explanations.
