Squares and Cubes

Squares and cubes are fascinating mathematical concepts that are all around us. They play a role in everyday life from building structures to understanding nature. Whether you are measuring the surface area and volume of your room, creating a work of art or exploring the mysteries of the universe, these mathematical concepts are used. Let's dive into the world of squares and cubes.

Square

→ A square is a value that you get when you multiply a number by itself. 
→ The square of a number x is x².

Example: Square of 8 is 64. This means 8 multiplied by 8 to give 64.

8² = 8 × 8 = 64

→ Square numbers are always positive.
→ Square of the numbers 1 to 30 is shown as:

Square Root

→ The square root of a number is the value that when multiplied by itself gives the original number.
→ The square root is represented by the symbol ‘√’. 

Example: Square root of 81 is 9 because 9 multiplied by itself is 81.
√81 = √(9 × 9) = 9

Perfect Square

→ If you can express a number (x) as the square of a whole number (x²), that number is called a perfect square.

Example: 121 is a perfect square number because it is a square of 11.

√121 = √(11 × 11) = 11

Properties of Squares

1. When you square the first ten natural numbers, the results end in digits 0, 1, 4, 5, 6 or 9. So, numbers ending in 2, 3, 7 or 8 are not perfect squares.
2. Squares of even numbers are always even.
3. Squares of odd numbers are always odd.
4. Perfect square numbers have an even number of zeros at the end.
5. Square numbers are always positive and never negative.

Pythagorean Triplets

→ Pythagorean triplets are sets of three natural numbers m, n and p
such that m² + n² = p².

→ Pythagorean triplets are given by the formula:
2m, m² + 1, m² − 1 

For any number m > 1
The members of Pythagorean triplets are 2m, m² + 1 and m² − 1.

Finding the Square Root

Let's simplify how to find the square root using two methods:

Method 1: Prime Factorisation Method

a) To find the square root of a perfect number, first write down its prime factors (the numbers that can multiply together to give that number).
b) Then, group these prime factors into pairs by taking one from each pair.
c) Multiply these prime factors together and you will get the square root of the number.

Example: The square root of 784 using the prime factorisation method is shown as:

Method 2: Long Division Method

a) Take the number you want to find the square root.
b) Group its digits in pairs and start from the right (one place).
c) Think of the largest number whose square is equal to or just less than the first group of digits. This number will be your starting divisor and quotient.
d) Subtract the product of the divisor and quotient from the first group of digits and bring down the next group to the right. This new setup becomes your new dividend.
e) Now, create a new divisor by adding the previous divisor to itself and adding a suitable digit to it. This digit should make the product of the new divisor and itself equal to or just less than the new dividend.
f) Repeat these steps for all the digit groups.
g) The final result you get is the square root of the given number.

Example: The square root of 784 using the division method is shown as:

Estimation Method

Imagine you have a number like 140 and you want to find its square root, but it is a bit tricky. You can estimate it using this method:

a) First, find a number that when squared is smaller than 140 (the lower limit).
b) Then, find a number that when squared is larger than 140 (the upper limit).
c) If you square 11 to get 121 (which is less than 140) and if you square 12 to get 144 (which is more than 140).
d) Hence, the square root of 140 lies between 11 and 12. To get a better estimate, you can start in the middle which is 11.5.
e) Since 12 squared is very close to 140, you can take a few smaller steps towards 11.8.
f) When you square 11.8, you will get around 139.24 which is very close to 140. So, you can estimate that the square root of 140 is approximately 11.8 (Round-off value).

This method helps you get a rough idea of the square root when you can't find it exactly.

Cube

→ A cube is a value that you get when you multiply a number by itself three times. 
→ The cube of a number x is x³.

Example: The cube of 4 is 64. This means 4 is multiplied thrice to give 64.

4² = 4 × 4 × 4 
= 64

→ Cube of the numbers 1 to 20 is shown as:

Cube Root

→ The cube root of a number is the value that when multiplied by itself three times gives the original number.
→ The cube root is represented by the symbol ‘³√’. 

Example: Cube root of 27 is 3 because 3 multiplied by itself thrice is 27.

³√27 = ³√(3 × 3 × 3)
= 3

Perfect Cube

→ If you can express a number (x) as the cube of a whole number (x³), that number is called a perfect cube.

Example: 125 is a perfect cube number because it is a cube of 5.

³√125 = ³√(5 × 5× 5)
= 5

Finding the Cube Root

Let's simplify how to find the cube root using two methods:

Method 1: Prime Factorisation Method

a) First break down the given number into its prime factors.
b) Now, group these prime factors into sets of three equal factors. If you have any leftover factors, leave them in a group by themselves.
c) Pick one factor from each group of three and multiply them together. That result is the cube root of the original number. 

Example: The cube root of 1331 using the prime factorisation method is shown as:

Method 2: Long Division Method

a) Start by making pairs of three-digit numbers from the right end (unit's place) of the given number.
b) Next find the largest number whose cube root is less than or equal to the given number, when a cubed number is less than or equal to the original number.
c) Subtract this number from the given number and write down the result as the new number.
d) Repeat the same process with this new number by making pairs of three digits and finding the largest number whose cube root is less than or equal to it.
e) Continue these steps until you have processed all the pairs of three digits. The final result you get is the cube root of the original number.

Example: The cube root of 1331 using the division method is shown as: