What is a Square Root of Number?
Square root of a number is a number that, when multiplied by itself, gives the original number. Every number has both positive and negative square roots. A radical sign ‘\( \sqrt{~~}\)’ is used to represent a square root and the number under the radical sign is known as radicand. A perfect square is a positive number that is obtained by multiplying a number by itself. An imperfect square is a number that cannot be obtained when a whole number is multiplied by itself.
Square Root of 1 to 20
The following table shows the square roots of 1 to 20. Let’s go through it and make an attempt to memorize them.
Calculating the Square Root of a Number
The square root of any number can be calculated by two different methods.
1. Prime Factorization:
As the name suggests, prime factorization is the process of writing a number as a product of prime factors. Prime numbers are the numbers that have only two factors, that is 1 and the number itself. We use the prime factorization method to find the square roots of perfect squares.
Example: Find the value of \( \sqrt{9}\).
Prime factorization of 9 are \( 3~\times~3\), that is, \( 9~=~3~\times~3\)
\( \Rightarrow ~9~=~3^2\)
\( \Rightarrow ~\sqrt{9}~=~\pm 3\)
So, the square roots of 9 are 3 and -3.
2. Long Division Method
The square roots of imperfect squares can be determined using the long division method. We will study about the long division method in higher grades.
Solved Examples
Example 1: Rosy wants to hand paint a plate which has a radius of \(\sqrt{15}\) centimeters. Will Rosy be able to paint the plate with 2 pints of paint, which is sufficient to cover 50 square centimeters?
Solution:
Radius of the plate \( r~=~\sqrt{15}\) cm
To find whether Rosy can paint the plate completely with 2 pints of paint, we need to find the area of the circular plate.
\( A~=~\pi~r^2\) [Write the formula]
\( A~=~3.14~\times~(\sqrt{15})^2\) [Substitute the values]
\( A~=~3.14~\times~15\) [Find the square of \( \sqrt{15}\)]
\( A~=~47.1\) [Multiply]
Therefore, the total area of the plate is \( 47.1~cm^2\).
It is given that 2 pints of paint can cover \( 50~cm^2\). So, Rosy can paint the plate completely with 2 pints of paint.
Example 2: The area of the cover of a square shaped notebook is 16 square inches. What is the length and width of the cover?
Solution:
Use the formula for the area of the square to find the dimension of top.
\(A~=~a^2\) [Write the formula for area of a square]
\(16~=~a^2\) [Substitute the values]
\(\sqrt{16}~=~\sqrt{a^2}\) [Apply square root both side]
\(4~=~a\) [Simplify]
Therefore, the length of each side of the top of a notebook is 4 inches.
Example 3: Find the value of the equation \(6\sqrt{5}~+~8\sqrt{10}\).
Solution:
\(6\sqrt{5}~+~8\sqrt{10}\) [Write the expression]
\(\Rightarrow ~6~\times~2.236~+~8~\times~3.162\) [Substitute \(\sqrt{5}~=~2.236\) and \(\sqrt{10}~=~3.162 \)]
\(\Rightarrow ~13.416~+~25.296\) [Multiply]
\(\Rightarrow ~38.712\) [Add]
So, \(6\sqrt{5}~+~8\sqrt{10}~=~38.712\).
There are 16 imperfect squares from 1 to 20. They are 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, and 20.
There are four perfect squares from 1 to 20 that are 1, 4, 9, and 16. Since the square roots of perfect square numbers are integers and can be written in p/q form, they are rational numbers. Hence, there are four rational numbers from root 1 to root 20.
The square root of numbers from 1 to 20 can be calculated by using two different methods, prime factorization and long division method.
No, 8 is not a perfect square because the value of the square root of 8 is non-terminating and non-repeating.