# Square Root and Cube Root

Square Root and Cube Root: In mathematics, the square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16. Every non-negative real number x has a unique non-negative square root, called the principal square root, which is denoted by √x, where the symbol √  is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by √9 = 3, because 32 = 3 ⋅ 3 = 9 and 3 is non-negative.

1.   When we multiply a number by itself three times, the product so obtained is called the perfect cube of that number.
2. There are only 10 perfect cubes from 1 to 1000.
3. Cubes of even numbers are even and those of odd numbers are odd
4. The cube of a negative number is always negative.
5. If the digit in one�fs place of a number is 0, 1, 4, 5, 6 or 9, then its cube will end in the same digit.
6. If the digit in one�fs place of a number is 2, then its cube will end in 8 and vice-versa.
7. If the digit in one�fs place of a number is 3, then its cube will end in 7 and vice-versa.
8. If the prime factor of a number can not be made into groups of 3, it is not a perfect square.
9. The symbol 3�� denotes the cube root of a number.

•  Square of a number.
If a natural number m can be expressed as n2 (where n is a natural number), then m is the square root or perfect square.
i.e. if m = n2          (m, n – natural numbers)
E.g. 81 = 3 × 3 × 3 × 3
= 32 × 32 = (3 × 3)2 = 92
Hence, 9 is the square root of 81.

•  Properties of Square Root.
Below is the table that has squares of numbers from 1 to 10. If we see the above results carefully, we can conclude that numbers ending with 0, 1, 4, 5, 6, or 9 at units place are perfect squares None of these end with 2, 3, 7 or 8, So numbers that end with 2, 3, 7, 8 are not perfect squares.
Thus, numbers like 122, 457, 183, 928 are not perfect squares.

• One’s digit in square of a number.
(1). The ones digit in the square of number can be determined if the ones digit of the number is known. (2). The number of zeros at the end of a perfect square is always even and double the number of zeros at the end of the number
E.g. Double zero {7002 = 490000} four zero (even)
(3). The square of an even number is always an even number and square of an odd number is always an odd number.
E.g. • Interesting patterns of Square Root.

Number between square numbers
There are ‘2a’ non perfect square numbers between the square of two Consecutive natural numbers n + (n + 1)
Between 22 = 4 & 32 = 9 → 5, 6, 7, 8
2 × 2 = 4 non square numbers
Between
32 = 9 & 42 = 16 → 10, 11, 12, 13, 14, 15
2 × 3 = 6 non square numbers. 