MathWithNaziaa : Squares and Square roots & Cube and cube roots

 Squares and Square roots

Square and square roots both concepts are opposite in nature to each other. Both are inverse operations in mathematics.  The square of a number is the number that needs to be multiplied by itself to the actual number, but square root is a number that needs to be multiplied by itself to get the original number. 

 if n is a number then its square is represented by n raised to the power 2,ie,, n² And its square root is expressed as n , Where Is called radical, the value under the root symbol is called radicand. 

Properties of square numbers:

• Square of 0 is 0

• Square of 1 is 1

• Square of positive numbers is always positive

• Square of negative numbers always positive

 Squares of negative numbers:

                      The square of the negative number gives a positive value.  if we multiply two negative numbers the result will always be positive.

                           (-)  X(-) = (+)

For example:

Square root of a number:

                  The square root of any number is the value which when multiplied by itself gives the original number. 

Methods to find the square root of a number:

                There are two methods to find the square root of a number,  they are

• Prime factorization method

• Long division method    

 Prime factorization method:

           The prime factors of a perfect square can be paired to find the square root. First we make pairs of the prime factors. We can take one factor of every pair as the product of these factors.

For example:

1. Find the square root of 1764 by using the prime factorization method?

2. Find the square root of 4356  using the prime factorization method ?

Long division method:

           The prime factorization method becomes little difficult for large numbers in that case we use long division method which consists of Two Steps

1. Pairing

2.  long division

The pairing of a number in a decimal number to find the square root by long division method is done from right to left, the whole number portion from left to right. 

For example:

1. Find the square root of 1225?

Therefore square root of 1225 is 35

2. Find the square root of 104976?

The four square root of 104976 is 324.            

Cube and cube roots 

Cube:  the cube of a number raised to the power 3, Or when a number is multiplied 3 X itself, we can say that the number has been a cube and the product is called a cube of the number. If a is a number then the cube of a is

For example :

Perfect cube: 

              A natural number is said to be a perfect cube, if it is the cube of some natural number. A natural number “n” is a perfect cube if it is the cube of some natural number ,    ie.. n= m3

For Example:

1. Find  the cube root of 216 

2) Find the cube root of 512

3) find the cube root of 4096

Properties of cube

• The cube of an even number is always even. For example, a cube of four will be 64,  which is also an even number.

•  if any number is odd then  its cube will always be odd.  For example, a cube of 3 we will be 27 which is also an odd number.

•  If any number is a negative number then it will always be negative. For example, a cube of -2 will be -8 .