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Square Root of 40
The square root of a number is always a pair of the positive and negative values of a certain number. The square root of 40 implies a number which when multiplied by itself results in 40. We will now calculate the square root of 40 by approximation method and long division method along with few interesting problems.
 Square root of 40: √40 = 6.3245
 Square of 40: 40² = 1600
What is The Square Root of 40?
 The square root of any number is the number when multiplied with itself results in the given number.
 Square root of 40 is expressed as √40 = 6.3245
 Square root of 40 in radical form = √40 = √(2 × 2 × 2 × 5) = 2√10
 Therefore, 40 is not a perfect square
Is Square Root of 40 Rational or Irrational?
 A number is defined as a rational number when it can be expressed in the p/q form where q ≠ 0.
 As the square root of 40 is a nonterminating and nonrepeating number. So, the square root of 40 cannot be represented in the form of p/q.
 Hence, the square root of 40 is an irrational number.
How to Find The Square Root of 40?
We can calculate the square root of 40 by approximation method or long division method.
Square Root of 40 by Approximation Method
 Firstly, we need to find two perfect squares between which 40 lies.
 We know that 36 (6^{2}) and 49 (7^{2}) are the two perfect squares between which 40 lies.
 So, the square root of 40 will be greater than 6 but less than 7.
6 < √40 < 7
Therefore, the whole number part will be 6.  For the decimal part we will use the formula:
Given number Lower perfect square / Bigger perfect squareLower perfect square
= (4036) / (4936) = 4/13 = 0.31
So, the approx. the square root of 40 will be 6.31
Square Root of 40 By Long Division
By following the below steps, we can find the square root of 40 by the long division method.
Step1. Write 40 as shown below in the diagram. Start pairing the digits from one’s place in pairs of two by putting a bar on top of them.
Step2. Now find a number such that when it is multiplied with itself results in a number less than equal to 40. We know that 6 × 6 = 36
Step3. Now subtract it from the dividend as done in the usual division and add the divisor with itself that was calculated in the previous step. The divisor will become 12 and the remainder will be 4.
Step4. As no more numbers are left in the dividend so, we put a decimal point after the dividend and quotient simultaneously. Now place three pairs of zeros after the decimal in the dividend part and bring the first pair of zeros down.
Step5. Now look for a number at the unit’s place of divisor such that it results in a number less than equal to 400. Here the number will be 3 as 123 × 3 = 369 (less than 400)
Step6. Bring the next pair of zeros down and repeat the steps till the last pair of zeros.
Therefore, we get the square root of √40 = 6.324 by the long division method.
Explore square roots using illustrations and interactive examples
Important Notes:
 The square root of 40 is expressed as √40 in radical form.
 There will be n/2 digits in the square root of an even number with n digits.
 There will be (n+2)/2 digits in the square root of an odd number with n digits
 The square root of 40 is written as (40)^{1/2} in exponential form.
Challenging Questions:
 By what number 40 should be divided to make it a perfect cube?
 Find the square root of all the factors of 40?
Square Root of 40 Solved Examples

Example 1: Find the square root of 40 using the prime factorization method?
Solution:
Prime factorization of 40: 2^{3} × 5
Prime factors of 40 in pairs: (2 × 2) × 2 × 5
Square root of 40: √((2 × 2) × 2 × 5) = √(2^{2} × 2 × 5)
Therefore, √40 = 2√10 
Example 2: David is painting a squareshaped table with a surface are 40 square feet and he wants to add a LED strip around the sides of the table. What length of LED strip does David need?
Solution:
Area of the table = (Length of side)^{2} = 40
Length of side = Square root of 40 = √40 = 2√10
Total length of LED strip required = 4 × Length of one side = 4 × 2√10 = 8√10
Hence, the required length of the LED strip = 8√10 feet.
FAQs on Square Roots of 40
What is the square root of 40 using prime factorization of 40?
Prime factorization of 40: 2^{3} × 5
√40 = √(2 × 2 × 2 × 5) = 2√10.
What are the square roots of 40?
The square roots of negative numbers are imaginary.
These are expressed as √40 = +2√10 i and 2√10 i.
Is the square root of 40 is a rational number?
No, the square root of 40 is not a rational number
Because the square root of 40 is a nonterminating and nonrepeating number. So, these cannot be represented in the form of p/q.
What is the negative square root of 40?
The negative square root of 40 is 2√10
As 2√10 × 2√10 = 40.
Can we find the square root of 40 by the repeated subtraction method?
No, we can’t find the square root of 40 by repeated subtraction method
As this method can only be used in the case of a given number is a perfect square and 40 is not a perfect square.
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