Squares are the numbers, generated after multiplying a value by itself. The square root of a number is a value that, when multiplied by itself, gives the number. For example, the square of 3 is 9 and the square root of 9 is 3.
If n is a number then its square is represented by n raised to the power 2, i.e., n2 and its square root is expressed as ‘√n’, where ‘√’ is called radical.
area of a square = side × side (where ‘side’ means ‘the length of a side of the Square’)
| Side of square (in cm) | Area of square ( in cm2 ) |
| 1 | 1 x 1= 1 |
| 2 | 2 x 2 = 4 |
| 3 | 3 x 3 = 9 |
| 5 | 5 x 5 = 25 |
| 8 | 8 x 8 = 64 |
| a | a x a = a^2 |
Properties of Square Numbers
- Square of 1 is equal to 1
- Square of positive numbers are positive in nature
- Square of negative numbers is also positive in nature. For example, (-3)2 = 9
- Square of zero is zero
- Square of root of a number is equal to the value under the root. For example, (√3)2 = 3
- The unit place of square of any even number will have an even number only.
- If a number has 1 or 9 in the unit’s place, then its square ends in 1.
- If a number has 4 or 6 in the unit’s place, then its square ends in 6.
Numbers between square numbers
- (n + 1)^2 – n^ 2 = (n^ 2 + 2n + 1) – n ^2 = 2n + 1
- We find that between n ^2 and (n + 1)^2 there are 2n numbers which is 1 less than the difference of two squares
- Thus, in general we can say that there are 2n non perfect square numbers between the squares of the numbers n and (n + 1)
Squares of Negative Numbers
The squares of negative numbers give a positive value, because if we multiply two negative numbers then it will result in a positive number.
Remember that: (-) x (-) = (+)
Therefore, square of (-n), (-n)^2 = (-n) x (-n) = n^2
Where n is a number.
Examples:
- (-5)2 = (-5) x (-5) = 25
- (-7)2 = (-7) x (-7) = 49
Square Roots of Number
The square root of any number is the value which when multiplied by itself gives the original number. It is denoted by the symbol, ‘√’. If the square root of n is a, then a multiplied by a is equal to n. It can be expressed as:
√n = a then a x a = n
This is the formula for square root.
Square Roots of Perfect Squares
The perfect squares are the one whose square root gives a whole number. For example, 4 is a perfect square because when we take the square root of 4, it is equal to 2, which is a whole number. Let us see some of the perfect squares and their square roots.
Square Root of Imperfect Squares
Finding the square root of perfect squares is easy but to find the root of imperfect squares is difficult. The root of the perfect square can be estimated using the prime factorisation method.
The square root of imperfect squares is usually fractions. For example, 2 is an imperfect square because 2 cannot be prime factorised and its square root gives a fractional value.
Examples are:
- √2 = 1.4142
- √3 = 1.7321
- √8 = 2.8284
QUESTIONS:
Example 1: Find the square root of 6400.
Solution: Write 6400 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5
Therefore 6400 = 2 × 2 × 2 × 2 × 5 = 80