Square Root of 10
Square Root
The square root of any number gives the same number when the number multiplied by itself.
For example – \[\sqrt{p\times p}\] = \[\sqrt {(p)^{2}}\] = p
Square Root Symbol
The symbol used to denote the square root is “√”. It is also known as a radical symbol or radix. The number written under the square root symbol is called the radicand. The square value can be represented in the radical form as well as in decimal form. Square root of 10 can also be represented as a radical of 10.
How to Calculate the Value of a Square Root of 10?
Calculating the root 10 value is a bit complex because the number 10 is not a perfect square as its unit digit is 0. Square root of a number can be easily obtained if the number is in a perfect square. The number is considered as a perfect square if it can be denoted as a product of two equal integers.
For example 5×5= 25, it is representing the square of a number 5. It is considered as a perfect square as it is stated as a product of two similar integers i.e. 5 x 5 = 25, 6 x 6= 36. It is representing the square of 6. It is even considered as a perfect square as it is stated as a product of two similar integers i.e. 6 x 6.
The number is a perfect square if the unit place of a number ends with 1,4,5,6 or 9
The number is not said to be a perfect square if it ends with 2, 3, 7 or 8.
We can calculate the value of a square root or root 10 values through two methods.

First method to calculate the value of a square root or root 10 value is to use the unit digit of the given number,

The second method to calculate the square root value or root 10 values of the given number is by using a long division method.
What is the Square Root of 10?
The square root of 10 or root 10 is represented in the form of √10. As we know 10 is an even number but not a prime number. Prime numbers are considered as those numbers which have only two factors i.e. 1 and the number itself. For example 2 is a prime number as it has only factors 1 and 2 itself. But number 10 is not a prime number because it has multiple factors like.
1×10 = 10
2 x 5= 10
10 x 1 = 10
5 x 2 = 10
To calculate the square root of value 10, write its factors first
10 = 2 x 5
Square root of 10 can be written in the below format
\[\sqrt 10\] = \[\sqrt 2\] X \[\sqrt 5\]
Common square terms out of the root in the above equation cannot be taken out as it has no common square terms.
\[\sqrt 10\] =\[\sqrt 2\] \[\sqrt 5\]
\[\sqrt 10\] or root 10 is in the radical form. If you want to write it in a decimal form, then substitute approximate values of \[\sqrt 2\] and \[\sqrt 5\] which is 1.414 and 2.236 respectively,
^{8}\[\sqrt 10\] =1.414 x 2.236
\[\sqrt 10\] =3.162
Hence, the value of root 10 or root 10 is 3.162
Square root of 10 using Long Division Method
(division method image will be uploaded soon)
Square Root Values
Solved Example
1. Find the value of √80 + 16√5 , if 3√5+√125 = 17.88
Solution: \[\frac{x}{\sqrt{512}}\] = \[\frac{\sqrt{648}}{x}\]
= 3√5 + √125 = 17.88
= 3√5 + (√25 x √5) = 17.88
= 3√5 + 5√5 = 17.88
= 8√5 = 17.88
= √5 = 17.88/8
= √80 + 16√5 = √16 x √5 + 16 √5
= 4√5 + 16√5 = 20√5
= 20 x 17.88/8 = 44.7
2. Simplify: (√7 1/√7)2
= (√7 1/√7)2
= (√7 1/√7)2
= (√7)2 – 2 x √7 x 1√7 + (1/√7)2
= 72 +1/7
= 5 + 1/7
= 36/7
Fun Facts

The Yale Babylonian has a tablet from nearly 4000 years ago that states the square root of 2 out to 9 decimal places by making use of a square and two diagonals.

Communities in Ancient India were making use of square roots since 800 BCE.

An Indian Mathematician from the 9th century named Mahavira is the first person to announce that negative square roots do not take place.

Procedure to determine the square root is outlined in the Chinese book, Writings on Reckoning, written in around 200 BCE during the Han Dynasty.
Quiz Time
1. A square garden having the area measurement of 225 square feet. How much fencing will be needed by the gardener to purchase to fix fencing around the garden?

60 ft

112.5 ft

15 ft

56.25 ft
2. What will be the length of one side of a square, if the area of a square is 100 meters?

√10

25

10

50
3. What will be the value of : \[\sqrt 0.001\] + \[\sqrt 0.81\] + \[\sqrt 1.21\] + \[\sqrt 0.0009\]

2.13

2.03

2.11

2.1
1. How to Find the Square Root of Large Numbers Using its Unit Digit?
Finding the square root of small numbers is an easy task as we are aware of the multiplication table of 1 to 10. The number, when multiplied by itself, gives the square in two digits. But it is difficult to find out the square root of three fourdigit numbers.
Here, we will learn an easy method to calculate the square root of a larger number 4489
Step 1. The unit digit of the number 4489 is 9, which implies that the unit digit of the square root of a number 4489 will be either 3 or 7 because 32 is 9 and 72 is 49.
Step 2. Now, we will take the first two digit of number 4489 i.e. 44 which lies between the squares of 6 and 7 because 62 ˂44˂72.
Step 3. We can assume that that the tens digit of a square root of a number 4489 is 6 as t is lowest among two number 6 and 7, Now, we need to calculate unit digit of the square root of the number 4489.
Step 4. Now the square root of a digit 4489 will be between 63 and 67.
Step 5. Since the tens digit is 6 and the next number is 7, we will multiply these both digit like 6 X7 = 42 as 42 is less than 44.
Step 6. The square root of 4489 will be the greater number between 63 and 67 and the number will be 67.
Hence,√ 4489 is 67.
2. What are the Applications of a Square Root?
The square root formula is the most significant in Mathematics as on the basis of the square root formula, square root problems given in Maths book can be solved. The basic application of square root is used in computing the square root problems. Square root values can be used to simplify problems. For Example, if the equation is given in radical form such as the square root of 4√4.It will be difficult to solve such an equation. Substituting the square root value will simplify the equations. Computation of square root numbers can also be done in personal calculators that have square root function uploaded.
Square root is also used in geometry where one can easily find the area of a square to its length. It is also used to find out the formulas for roots of quadratic integers, quadratic fields and quadratic integers.