Square root of 40

The square root of 40 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y × y, the result is 40. It contains both positive and a negative root, where the positive root is called the principal square root. The square root of 40 is ±6.32455532034. The positive value, 6.32455532034 is the solution of the equation x² = 40.

As defined, the square root is just the inverse of squaring a number, so, squaring 6.32455532034 will result in 40.

The square root of 40 is expressed as √40 in radical form, where the ‘√’  sign is called the “radical”  sign. In exponential form, it is written as (40)^1/2  
 

We can find the square root of 40 through various methods. They are:

i) Prime factorization method

ii) Long division method

iii) Approximation/Estimation method
 

The prime factorization of 40 involves breaking down a number into its factors. Divide 40 by prime numbers, and continue to divide the quotients until they can’t be separated anymore. After factoring 40, 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 40 = 2 × 2  × 2  × 5 

for 40, only one pair of factors 2 can be obtained, but a single 2 and 5 are remaining.

So, it can be expressed as  √40 = √(2 × 2  ×2 ×5) = 2√10

2√10 is the simplest radical form of √40.
 

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 40:

Step 1: Write the number 40, 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 40. Here, it is
             6, Because 6²=36< 40.

Step 3 : Now divide 40 by 6 (the number we got from Step 2) such that we get 6 as quotient and we get a remainder. Double the divisor 6, we get 12, and then the largest possible number A1=3 is chosen such that when 3 is written beside the new divisor, 12, a 3-digit number is formed →123, and multiplying 3 with 123 gives 369 which is less than 400.

Repeat the process until you reach the remainder of 0

 We are left with the remainder, 7024 (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 6.324….

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 40. Here, it is 36 and 49.

Step 2: We know that, √36=±6 and √49=±7. This implies that √40 lies between 6 and 7.

Step 3: Now we need to check √40 is closer to 6 or 6.5. Since (6)²=36 and (6.5)²=42.25. Thus, √40 lies between 6 and 6.5.

Step 4: Again considering precisely, we see that  √40 lies close to (6.5)²=42.25. Find squares of (6.2)²=38.44 and (6.4)²= 40.96.

We can iterate the process and check between the squares of 6.3 and 6.39 and so on.

We observe that √40 = 6.324…
 

• 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. Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. 

• Perfect and non-perfect square numbers - Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :2, 8, 18