Square Root of 7
Introduction:
Square root of a number is a value obtained by raising the number to the power ½. It is denoted by the symbol √ . Square root of a number x is written as √x or x½. Square root of a number and square of a number are the inverse operations in Mathematics. Any number when multiplied by itself gives the square of a number. Square root of 7 can be calculated using various methods. The most precise method of calculating the square root of 7 is the long division method.
Determination of Square Roots:
Before finding the square root of any number, it is important to check whether the number is a perfect square or not. All numbers are not perfect square numbers. The numbers which have its square root as integers are called the perfect square numbers. To find the square root of perfect square numbers, any one of the following methods can be used.

Prime Factorization method

Repeated subtraction method

Long division method

Number line method

Average method

Binomial expansion method

Calculus method
However, if the number is not a perfect square number prime factorization method and repeated subtraction method cannot be used.
How to Find the Square Root of 7 by Average Method?
To determine the value of root 7 using the average method, the following steps are to be executed.
Step 1:
Find out the two perfect square numbers which are very close to the given number on either side.
For the number ‘7’, the immediate perfect square lesser than 7 is ‘4’ and the immediate perfect square greater than 7 is ‘9’.
Step 2:
Note down the square roots of the perfect squares identified in Step 1.
Square root of ‘4’ is ‘2’ and the square root of ‘9’ is ‘3’.
Step 3:
It can be inferred that the square root of a given number lies between the square roots of numbers determined in step 2.
Square root of ‘7’ is any number between 2 and 3.
Step 4:
Divide the number whose square root is determined by any of the numbers obtained in Step 2.
‘7’ can be divided either by ‘2’ or ‘3’.
Let us divide ‘7’ by ‘3’
\[\frac{7}{3} = 2.333\]
Step 5:
Find the average of the quotient and divisor in Step 4.
The average of 3 and 2.33 is to be found
\[Average = \frac{{3 + 2.33}}{2} = \frac{{5.33}}{2} = 2.665\]
Step 6:
To get an accurate value of the square root, an average of the answer in step 5 and divisor of step 4 can be found. Finding of average can be continued as many times as required to get a precise value.
The average of 3 and 2.665 is to be found if the answer in step 5 for the value of under root 7 is not precise.
However the answer in step 5 is almost the same as the value of under root 7 calculated using the calculator (2.646).
\[\left( {\sqrt 7 } \right)\]=2.665 by average method.
How to Find Square Root of 7 by Number Line Method:
To find the square root of any number using a number line, a clear understanding of the basic concept of “Pythagorean Theorem” is required.
Pythagorean theorem states that “In a right triangle, the square on the longest side (hypotenuse) is equal to the sum of the squares on the other two sides (base and perpendicular)”.
A series of steps to be followed to determine the root 7 value using a number line is explained below.
Step 1:
Construct a number line with a minimum two units on either side.
Step 2:
Label the unit 1 as A. From the point ‘A’ draw a perpendicular AB of unit length.
Step 3:
Join OB. According to Pythagorean theorem, the length of OB is calculated as
\[OB\sqrt {O{A^2} + AB} = \sqrt {{1^2} + {1^2}} = \sqrt 2 {\text{ }}units\]
Step 4:
From the point ‘B’ draw a perpendicular BC of 1 unit to the line OB. Join OC. The length of OC is calculated using pythagoras theorem as
\[OC = \sqrt {O{B^2} + B{C^2}} = \sqrt {{{\left( {\sqrt 2 } \right)}^2} + {1^2}} = \sqrt {2 + 1} = \sqrt 3 units\]
Step 5:
From the point C, draw a perpendicular CD of length two units. Join OD
The length of OD is determined using pythagorean theorem as
\[OD = \sqrt {O{C^2} + C{D^2}} = \sqrt {{{\left( {\sqrt 3 } \right)}^2} + {3^2}} = \sqrt {3 + 4} = \sqrt 7 {\text{ }}units\]
The length of OD gives the measure of square root of 7.
How to Find the Square Root of 7 by Long Division Method?
Long division method is the most convenient and easiest method to find the square root of any number. It gives an accurate value of any number. For a better understanding of ‘How to find the square root of 7 by Long Division method?’
Step 1:
‘7’ is represented as 7.000000 to determine the root 7 value upto 3 decimal places. Group the digits after the decimal point in pairs as 5. 00 00 00
Step 2:
Choose a perfect square whose value is below ‘7’. The perfect square below 7 is ‘4’ and its square root is ‘2’.
Step 3:
Represent that ‘2 times 2 gives 4’ by writing 2 in the place of dividend and quotient. The number ‘4’ is written below ‘7’. ‘4’ is subtracted from ‘7’ and the difference is 3.
Step 4:
The first pair of zeroes in the dividend is carried down and a decimal point is placed in the quotient.
Step 5:
Add 2 to the divisor. The sum will be 4. Now take a number succeeding 4 to get a 2 digit number such that when the two digit number is multiplied by the number taken, the product is less than 300. If we take the digit as ‘4’, 46 \[ \times \] 6 = 276 which is less than 300. So now the remainder is ‘24’ and the next divisor is 52_. And the dividend is 2400 if the next pair of zeroes are taken down.
Step 6:
If the above steps are continued, the final answer obtained in the place of quotient is 2.645.
Finding the Square Root of 7 Using Binomial Expansion:
Binomial expansion states that
\[{\left( {1 + x} \right)^n} = 1{ + ^n}{C_1}x{ + ^n}{C_2}{x^2}{ + ^n}{C_3}{x^3} + ……{x^n}\]
Square root 7 value is evaluated in binomial expansion as 7.645 using x = \[\frac{1}{{64}}\] However, the calculations are a bit complex.
Fun Facts:

Square root of 7 can also be evaluated using concepts in calculus.

The number to be added to 7 to make it a perfect square is 2 and the number to be subtracted from it to make it a perfect square in 3.
1. What are square roots? How are they useful?
Square root of a number is that number which when multiplied by itself gives the number whose square root is to be determined as an answer. Finding the square root of a number is equivalent to raising the same number to the power ½. Square root of a number ‘x’ can therefore be written in exponential form as x½. Computation of square and square roots in Mathematics are inverse operations. Square roots are used in solving quadratic equations and other complex Mathematical and Scientific calculations. Square root of a perfect square number is an integer whereas the square root of other numbers except perfect square numbers are decimal numbers. Square root of a number has two values i.e. positive and negative with the same magnitude.
2. How is the square root 7 value calculated?
‘7’ is not a perfect square number and it is a prime number. So, prime factorization and repeated subtraction methods cannot be used to compute the square root of 7. However, the value of square root of 7 can be derived by other mathematical calculations such as:

Average method

Calculus method

Long division method

Number line method

Binomial expansion method
Among all these above mentioned methods, the long division method is the most precise and convenient method of computing the square root of any number. The value of root 7 upto 3 decimal places calculated using the long division method is 2.645.