127 LearnersLast updated on May 26th, 2025
Square Root of 610
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in fields like vehicle design, finance, etc. Here, we will discuss the square root of 610.
What is the Square Root of 610?
The square root is the inverse of the square of a number. 610 is not a perfect square. The square root of 610 is expressed in both radical and exponential form. In the radical form, it is expressed as √610, whereas (610)^(1/2) in the exponential form. √610 ≈ 24.69818, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.
Finding the Square Root of 610
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, methods like the long division method and approximation method are used. Let us now learn the following methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 610 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 610 is broken down into its prime factors:
Step 1: Finding the prime factors of 610 Breaking it down, we get 2 x 5 x 61: 2^1 x 5^1 x 61^1
Step 2: Now we found out the prime factors of 610. The second step is to make pairs of those prime factors. Since 610 is not a perfect square, the digits of the number cannot be grouped in pairs. Therefore, calculating 610 using prime factorization is impossible.
Square Root of 610 by Long Division Method
The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.
Step 1: To begin with, we need to group the numbers from right to left. In the case of 610, we need to group it as 10 and 6.
Step 2: Now we need to find n whose square is 6. We can say n as ‘2’ because 2 x 2 = 4 is lesser than or equal to 6. Now the quotient is 2, and after subtracting 4 from 6, the remainder is 2.
Step 3: Now let us bring down 10, which is the new dividend. Add the old divisor with the same number 2 + 2, we get 4, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 4n as the new divisor, we need to find the value of n.
Step 5: The next step is finding 4n × n ≤ 210. Let us consider n as 5, now 4 x 5 x 5 = 100.
Step 6: Subtract 210 from 100, the difference is 110, and the quotient is 25.
Step 7: Since the dividend is greater than the divisor, we need to continue. Now the new dividend is 11000.
Step 8: Now we need to find the new divisor that is 49 because 495 x 9 = 4455.
Step 9: Subtracting 4455 from 11000, we get the result 6545.
Step 10: Now the quotient is 24.69.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero. So the square root of √610 is approximately 24.698.
Square Root of 610 by Approximation Method
The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 610 using the approximation method.
Step 1: Now we have to find the closest perfect square of √610. The smallest perfect square less than 610 is 576, and the largest perfect square more than 610 is 625. √610 falls somewhere between 24 and 25.
Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (610 - 576) ÷ (625 - 576) = 0.68. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 24 + 0.68 = 24.68, so the square root of 610 is approximately 24.68.
Common Mistakes and How to Avoid Them in the Square Root of 610
Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in long division methods. Now let us look at a few of those mistakes that students tend to make in detail.
Mistake 1
Forgetting about the negative square root
It is important to make students aware that a number does have both positive and negative square roots. However, we will be taking only the positive square root, as it is the required one.
For example: √50 ≈ 7.07, there is also -7.07 which should not be forgotten.
Mistake 2
Not adding square root symbol in proper places
Not using the square root properly is another mistake that children make. To resolve this, we need to teach children that simplifying the numbers inside the square root is important and then move on to the next step.
For example: √(12+24) = √12 + √24 is the wrong method of using roots. The right way of using roots is √(12+24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to children is to simplify the numbers inside the square root. This is important so that students do not make mistakes while identifying the approximate value.
For example: √50 ≈ 7 is wrong, the correct answer is √50 ≈ 7.071067.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between cube root and square root constantly as this will help them avoid mistakes.
For example: √50 and ∛50 are different.
Mistake 5
Making mistakes in long division
Finding a square root using the long division method involves many steps. Due to this, many students might skip a step accidentally, which results in a wrong answer at the end of the sum. This may either happen in subtracting or elsewhere.
Square Root of 610 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √610?
The area of the square is approximately 610 square units.
Explanation
The area of the square = side^2.
The side length is given as √610.
Area of the square = side^2 = √610 x √610 = 610.
Therefore, the area of the square box is approximately 610 square units.
Problem 2
A square-shaped building measuring 610 square feet is built; if each of the sides is √610, what will be the square feet of half of the building?
305 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 610 by 2 = we get 305.
So half of the building measures 305 square feet.
Problem 3
Calculate √610 x 3.
Approximately 74.094
Explanation
The first step is to find the square root of 610, which is approximately 24.698.
The second step is to multiply 24.698 with 3.
So 24.698 x 3 ≈ 74.094.
Problem 4
What will be the square root of (600 + 10)?
The square root is approximately 24.698
Explanation
To find the square root, we need to find the sum of (600 + 10).
600 + 10 = 610, and then √610 ≈ 24.698.
Therefore, the square root of (600 + 10) is approximately ±24.698.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √610 units and the width ‘w’ is 20 units.
We find the perimeter of the rectangle as approximately 89.396 units.
Explanation
Perimeter of the rectangle = 2 × (length + width). Perimeter = 2 × (√610 + 20) = 2 × (24.698 + 20) = 2 × 44.698 = 89.396 units.
FAQ on Square Root of 610
1.What is √610 in its simplest form?
2.Mention the factors of 610.
3.Calculate the square of 610.
4.Is 610 a prime number?
5.610 is divisible by?
6.How does learning Algebra help students in Singapore make better decisions in daily life?
7.How can cultural or local activities in Singapore support learning Algebra topics such as Square Root of 610?
8.How do technology and digital tools in Singapore support learning Algebra and Square Root of 610?
9.Does learning Algebra support future career opportunities for students in Singapore?
Important Glossaries for the Square Root of 610
- Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, that is √16 = 4.
- Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero, and p and q are integers.
- Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
- Prime factorization: The expression of a number as the product of its prime factors. For example, the prime factorization of 610 is 2 x 5 x 61.
- Long division method: A method used to find the square root of non-perfect squares by dividing the number into smaller parts, making it easier to estimate the square root.
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Jaskaran Singh Saluja
About the Author
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.
Fun Fact
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