Square root of 7

The square root of 7 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y × y, the result is 7. It contains both positive and a negative root, where the positive root is called the principal square root. The square root of 7 is ±2.645.
The positive value, 2.645 is the solution of the equation x² = 7. As defined, the square root is just the inverse of squaring a number, so, squaring 2.645 will result in 7.  The square root of 7 is expressed as √7 in radical form, where the ‘√’  sign is called “radical”  sign. In exponential form, it is written as (7)^1/2  .

We can find the square root of 7 through various methods. They are:

i) Prime factorization method

ii) Long division method

iii) Approximation/Estimation method

The prime factorization of 7 involves breaking down a number into its factors. Divide 7 by prime numbers, and continue to divide the quotients until they can’t be separated anymore. After factorizing 7, make pairs out of the factors to get the square root.

If there exists numbers which cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs

So, Prime factorization of 7 = 7 × 1  

for 7, no pairs of factors can be obtained, only a single 7 is there.

So, it can be expressed as  √7 = √(7 × 1) = √7

√7 is the simplest radical form of √7.

This is a method used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.

Follow the steps to calculate the square root of 7:

Step 1: Write the number 7, and draw a bar above the pair of digits from right to left.
               
Step 2: Now, find the greatest number whose square is less than or equal to 7. Here, it is 2, Because 2²=4< 7.

Step 3 : Now divide 7 by 2 (the number we got from Step 2) such that we get 2 as quotient  and we get a remainder. Double the divisor 2, we get 4, and then the largest possible number A1=6 is chosen such that when 6 is written beside the new divisor, 4, a 2-digit number is formed →46, and multiplying 6 with 46 gives 276 which is less than 300.

Repeat the process until you reach the remainder of 0. We are left with the remainder, 3975 (refer to the picture), after some iterations and keeping the division till here, at this point. 

             
Step 4 : The quotient obtained is the square root. In this case, it is 2.645….
 

Estimation of square root is not the exact square root, but it is an estimate, or you can consider it as a guess.

Follow the steps below:

Step 1: Find the nearest perfect square number to 7. Here, it is 4 and 9.

Step 2: We know that, √4=±2 and √9=±3. This implies that √7 lies between 2 and 3.

Step 3: Now we need to check √7 is closer to 2.5 or 3. Since (2.5)²=6.25 and (3)²=9. Thus, √7 lies between 2.5 and 3.

Step 4: Again considering precisely, we see that  √7 lies close to (2.5)²=6.25. Find squares of (2.6)²=6.76 and (2.7)²= 7.29.

We can iterate the process and check between the squares of 2.61 and 2.69 and so on.

We observe that √7 = 2.645

• Exponential form - An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent. Ex: 3 ⤬ 3 ⤬ 3 ⤬ 3 = 81 Or, 3 ⁴ = 81, where 3 is the base, 4 is the exponent 

• Factorization - Expressing the given expression as a product of its factors Ex: 52=2 ⤬ 2 ⤬ 13 

•  Prime Numbers - Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....

•  Rational numbers and Irrational numbers - The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers.