Cubes and Cube Roots

Cubes and Cube Roots Class 8 Notes given here come in an easy to understand format and students can quickly go through all the topics given in chapter 7. The notes will help students to understand and remember the concepts better and prepare well for the exams.

Cube Numbers

Cube Numbers

If a natural number m can be expressed as $n^3$, where n is also a natural number, then m is called the cube number of n.
Numbers like 1, 8, 27  are cube number of the numbers 1, 2, and 3 respectively.
All perfect cube numbers are obtained by multiplying a number by itself three times.

Cubes Relation with Cube Numbers

In geometry, a cube is a solid figure where all edges are equal and are perpendicular to each other.

For example, take a cube of unit side. If we arrange these cubes to form a bigger cube of side 3 units, we find that there are a total of 27 such unit cubes that make up a cube of 3 units. Similarly, a cube of 4 units will have 64 such unit cubes.

Units Digits in Cube Numbers

Depending on whether a number is odd or even, its cube number is also odd or even respectively.
This is determined by the nature of the cube numbers’ unit digit.

• If a number is odd, its cube numbers’ unit digit is also odd.
• If a number is even, its cube numbers’ unit digit is also even.

The table below shows the units digit of a number and the units digit of the cube of that number:

Units digit of number	Units digit of its cube
1	1
2	8
3	7
4	4
5	5
6	6
7	3
8	2
9	9

Inside Cube Numbers

Adding Consecutive Odd Numbers

1 = 1 = 13

3 + 5 = 8 = 23

7 + 9 + 11 = 27 = 33

13 + 15 + 17 + 19 = 64 = 43

21 + 23 + 25 + 27 + 29 = 125 = 53
We can see from the above pattern, if we need to find the $n^3$, $n$ consective odd numbers will be needed, such that their sum is equal to $n^3$.
This pattern holds true for all natural numbers.
Also, if we need to find $n^3$ then we should add $n$ consecutive natural numbers starting from [latex](\frac{\frac{(n-1)(n)}{2}}{1})^{th}[/latex] odd natural number.

Prime Factorisation Method to Find a Cube

In the prime factorisation of any number, if each prime factor appears three times, then the number is a perfect cube.

Consider, the number 216. By prime factorisation,
$216=2\times 2\times 2\times 3\times 3\times 3=2^3\times 3^3=6^3$
Hence, 216 is a perfect cube.

Consider, the number 500. By prime factorisation,
$500=2\times 2\times 5\times 5\times 5=2^2\times 5^3$
In the above prime factorisation $2$ appears twice.
Hence, 500 is not a perfect cube.

To know more about Finding the Cube of a Number, visit here.

Cube Roots

Finding the cube root is the inverse operation of finding the cube.
We know that $3^3=27$. We can also write the same equation as $\sqrt[3]{327}=3$. The symbol ‘$\sqrt[3]{3}$‘ denotes ‘cube root‘.

Cube root using prime factorisation

We can find the cube root of a number by prime factorisation method by the following steps:

• resolve the number into its prime factors. Consider the number 5832. $5832=(2\times 2\times 2)\times (3\times 3\times 3)\times (3\times 3\times 3)$.
• make groups of three same prime factors.
• take one prime factor from each group and multiply them. Their product is the required cube root.

Therefore, cube root of $5832=\sqrt[3]{35832}=2\times 3\times 3=18$
To know more about Cube Roots, visit here.

Be More Curious!

Cube Root of a Cube Number using estimation

If a cube number is given we can find out its cube root using the following steps:

• Take any cube number say 117649 and start making groups of three starting from the rightmost digit of the number. So 117649 has two groups, and first group(649) and the second group(117).
• The unit’s digit of the first group (649) will decide the unit digit of the cube root. Since the number 649 ends with 9, the cube roots unit’s digit is 9.
• Find the cube of numbers between which the second group lies. The other group is 117. We know that $4^3=64$ and$5^3=125$. 64 < 117 < 125. Take the smaller number between 4 and 5 as the ten’s digit of the cube root. So, 49 is the cube root of 117649.

Differences of Squares of Triangular Numbers and Converse

Triangular numbers: It is a sequence of the numbers 1, 3, 6, 10, 15 etc. It is obtained by continued summation of the natural numbers. The dot pattern of a triangular number can be arranged as triangles.
​​​​​​
Sum of two consecutive triangular numbers gives us a square number. For example, $1+3=4=2^2$ and $3+6=9=3^2$.

The difference between the squares of two consecutive triangular numbers is a cube number. For example, $3^2-1^2=9-1=8=2^3$ and $6^2-3^2=36-9=25=5^3$.

Also, if the difference between the squares of two numbers is a cube number, then these numbers are consecutive triangular numbers.