364 LearnersLast updated on May 26th, 2025
Square root of 40
Square root is one of the most interesting mathematical topics to study. In daily life, square root functions are used in the field of engineering, GPS or distance calculations. Children use different approaches to solve square root problems. In this article, properties of square roots will be discussed.
What Is the 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
Finding the Square Root of 40
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
Square Root of 40 By Prime Factorization 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.
Square Root of 40 by Long Division Method
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….
Square Root of 40 by Estimation Method
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…
Common Mistakes and How to Avoid Them in the Square Root of 40
When we find the square root of 40, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.
Mistake 1
Assuming √40 as a simple fraction
√40=6.324… which is an irrational number. Hence, it cannot be expressed as a fraction
Mistake 2
Ignoring precision
Provide an approximate answer without acknowledging the extent of precision is not expected
Mistake 3
Assuming √40 as a perfect square
firstly find the approximate value of the square root before assuming it as a perfect square.
Mistake 4
Incorrect formula for finding square roots through different methods
The formula should be applied accurately for iterative methods
Mistake 5
Misunderstanding how to calculate the square root
To get a correct value of square root of 40, one should use methods like the Long Division Method to estimate.
Square Root of 40 Examples
Problem 1
Simplify √40 + 5√40 + √40?
√40 + 5√40 + √40
= √40(1+5+1)
= 7√40 = 7⤬2√10
= 14√10
Explanation
Simplified the expression and found out the simplest radical form of √40 that is 2√10, applied that and solved.
Problem 2
What is √40 multiplied by 2√40 and then divided by (√40)²?
(√40 ⤬ 2√40)/(√40)2
= (40⤬2)/40
= 2
Answer: 2
Explanation
√40 multiplying with itself gives 40, and then again multiplied by 2 in the first step and then again divided by (√40)2=40
Problem 3
Find the value of (1/√40)⤬ (1/√40)?
(1/√40)⤬ (1/√40) = 1/ 40 = 0.025
Answer: 0.025
Explanation
(1/√40)⤬ (1/√40) = 1/ 40 as same as √40⤬ √40 =40
Problem 4
If y=√40, find y², y³
firstly, y=√40
Now, squaring y, we get,
y2= (√40)2=40
or, y2=40
Similarly, y3=(√40)3=40√40
Answer : 40,40√40
Explanation
squaring “y” which is same as squaring the value of √40 resulted to 40 and hence applied this fact to each problem here.
Problem 5
Find √40 / √49
√40/√49
= √(40/49)
= 5.477/7
= 0.782…
Answer : 0.782…
Explanation
finding out the square root values of √40 and √49 and then dividing.
FAQs on 40 Square Root
1.How can you express √40 in fractional form ?
2.What is the value of √42?
3.Is 40 a perfect square or non-perfect square?
4.Is the square root of 40 a rational or irrational number?
5.How to solve √37?
6.How does learning Algebra help students in Saudi Arabia make better decisions in daily life?
7.How can cultural or local activities in Saudi Arabia support learning Algebra topics such as Square root of 40?
8.How do technology and digital tools in Saudi Arabia support learning Algebra and Square root of 40?
9.Does learning Algebra support future career opportunities for students in Saudi Arabia?
Important Glossaries for Square Root of 40
- 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
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Jaskaran Singh Saluja
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Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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