Square Root of 7

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.

Step-by-Step Explanation of the Square Root of 7 is Available at Vedantu.

As you may already know that 7 is a prime number, and it does not have perfect squares, therefore understanding the method to find the square root of such numbers, that is to say, the number which does not have a perfect square is very much important and extremely useful for the students. Because normally when we multiply a particular number by itself, then that particular number becomes the square root of its product, for example when we multiply 2 by 2 it results in 4 and hence 2 is the square root of the 4. But such is not the case with 7, because its square root is 2.646, but understanding how 2.646 is derived is of utmost importance.

Therefore, to make it easy for the students to find the square root of the 7, Vedantu provides a complete step-by-step explanation of the same. And also, this explanation of the method of finding the square root of 7 is designed in such a manner that understanding the same becomes extremely easy for the students, and more importantly, it is completely free of cost.

Overview of the Square in Mathematics.

Before understanding the method of finding the square of the number 7, it is important to have a brief idea or rather revise the concept of square root in itself. So, as said earlier when one number is multiplied by itself then it becomes the square root of its product. Let us take an example, if we multiply the number 5 with itself, we have 25, which in mathematics can be expressed in the following manner:

5 × 5 = 25

52 = 25

Therefore, the square of the number 5 is 25, while 5 is the square root of 25. The square root is represented by this symbol √, which is known as a sign radical or simply a radix. And so the square root of 5 is expressed by using the radix is √25, while the sign of the square root is considered as radix, the number inside of it is called radicand, and hence it the given example 25 is the radicand as it is expressed under the radix or the radical sign.

One thing that the students must keep in mind is that 5 is not the only square root of the 25, because it has another square root as well, which is -5. Yes, the square of 5 (positive number) and -5 (negative number) is 25, because when we multiply -5 by itself then its product becomes a positive number which is to say 25, and hence -5 and 5 are both the square roots of the 25.

Therefore, each and every number which is a positive number consists of 2 square roots, one positive, and the other negative. For instance, 4 and -4 are the square roots of 16, and 6 and -6 are square roots of 36. And the same goes for the number seven, which is discussed in a detailed manner in the given article.

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 the 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).   

7 = 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’ draws 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

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 

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.