When we multiply the number by itself then the product is called the square of that number. For example, 5 × 5 = 25, square of 5 is 25. We denote square of a number by writing ‘2’ as superscript of the number as 52. We can also say it as, ‘5 to the power of 2’.
Square of 5 = 52 = 5 × 5 = 25
Square of 6 = 62= 6 × 6 = 36
Square root of a number is just the opposite of square. To find the square root of x, we need to find a number, let's say 'a' whose square is x, i.e. a2=x. We can say square root of x is 'a'.
Square of 5 is 52 = 25
Square root of 25 is
\(\sqrt{25} = \sqrt{5\times5} = 5\)
Square root of 36 is
\(\sqrt{36} = \sqrt{6\times6} = 6\)
Note: Squaring a negative number gives positive result, -5 × -5 = +25, therefore square root of 25 is both +5 and -5. In mathematics, the square root of a number b is a number x such that x2 = b. For example, 3 and -3 are square roots of 9. This is because 32 or (-3)2 equals 9. principal square root is the positive number square root. These are denoted by √a where √ is referred to as the radix or the radical sign. For example, the principal square root of 16 is 4 which is denoted by √16 = 4, due to the fact that 42 = 4 x 4 = 16 and 4 is nonnegative. The number or term whose square root is being considered is referred to as the radicand. The radicand can also be described as the expression or number that is underneath the radical sign. In the example above, the radicand is 16.
Natural numbers that are squares of other natural numbers are called perfect square or square number. Following method can be used to find if a given number is a perfect square or not:
Find the prime factorization of the number and make pairs of equal factors. If all the factors can form pairs then it is a perfect square. For example,
120 = 2 × 2 × 2 × 3 × 5
Since all prime factors cannot be paired 120 is not a perfect square.
Let’s take another example – Find if 1296 is a perfect square
Prime factorization of 1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3
As all the factors can be paired so 1296 is a perfect square.
The square root of a perfect square can be determined using similar way
\(\sqrt{1296} = \sqrt{2\times2\times2\times2\times3\times3\times3\times3} = 2\times2\times3\times3\) (taking one factor from each pair)
Numbers that do not have perfect squares are called irrational numbers.
The cube of a number n is its third power, that is, the result of multiplying three instances of n together( n × n × n = n3). For example, cube of 3 is 27 ( 3×3×3). When 3 is cubed you get 27.
Cube of a number is three times multiplied by itself.
Cube of 2 = 2 × 2 × 2 = 8, we can say '2 to the power of 3 is 8'.
Cube of 5 = 53 =5 × 5 × 5 = 125
Cube of 6 = 63 = 6 × 6 × 6 = 216
Find the prime factorization of the number. If all the prime factors can be grouped into triplets of equal factors then the number is a perfect cube. Example–
1331 = 11 × 11 × 11
As equal factors can be grouped as triplets, it is a perfect cube. Let’s take another example 2916 = 3 × 3 × 3 × 3 × 3 × 3 × 2 × 2
since all factors cannot be grouped in triplets, 2916 is not a perfect cube.
