Squares and Square Roots

Squares and square roots both concepts are opposite in nature to each other. Squares are the numbers, generated after multiplying a value by itself. Whereas square root of a number is value which on getting multiplied by itself gives the original value. Hence, both are vice-versa methods. For example, the square of 2 is 4 and the square root of 4 is 2.

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. The value under the root symbol is said to be radicand.

The square numbers are widely explained in terms of area of a square shape. The shape of a square is such that it has all its sides equal. Therefore, area of square is equal to (side x side) or side². Hence, if the side length of the square is 3cm then its area is 3²= 9 sq.cm.

Properties of Square Numbers

The square numbers are the values which are produced when we multiply a number by itself. Some of the properties are:

• 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.

Also, read:

Square Numbers 1 to 50

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) = n²

Where n is a number.

Examples:

• (-5)² = (-5) x (-5) = 25
• (-7)² = (-7) x (-7) = 49

Numbers between Squares

Suppose there are two square numbers n² and (n+1)², then total numbers between these two squares are given by 2n.

Let’s say 3² and 4² are two squares.

3² = 9 and 4² = 16

We need to find the numbers present between 9 and 16.

Here, n = 3

Therefore, total numbers between 9 and 16 = 2n = 2 x 3 = 16

Is that correct? Let us check.

9, 10, 11, 12, 13, 14, 15, 16.

As we can see, the numbers between 9 and 19 are 6. Therefore, the formula given above is applicable to all the squares.

Square Roots of Number

As we have already discussed, 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

Video Lessons on Square Roots

Visualising square roots

Finding Square roots

Square Roots 1 to 50