118 LearnersLast updated on May 26th, 2025
Square Root of 1537
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 the fields of vehicle design, finance, etc. Here, we will discuss the square root of 1537.
What is the Square Root of 1537?
The square root is the inverse of the square of the number. 1537 is not a perfect square. The square root of 1537 is expressed in both radical and exponential form. In the radical form, it is expressed as √1537, whereas 1537^(1/2) in exponential form. √1537 = 39.197, 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 1537
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where 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 1537 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 1537 is broken down into its prime factors.
Step 1: Finding the prime factors of 1537 Breaking it down, we find that 1537 is a product of smaller prime numbers: 13 x 118, 13 x 2 x 59.
Step 2: Now we found out the prime factors of 1537. The second step is to make pairs of those prime factors. Since 1537 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 1537 using prime factorization is not feasible.
Square Root of 1537 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 1537, we need to group it as 37 and 15.
Step 2: Now we need to find n whose square is less than or equal to 15. We can say n is 3 because 3 x 3 = 9 is less than 15. Now the quotient is 3, and after subtracting 9 from 15, the remainder is 6.
Step 3: Now let us bring down 37, which is the new dividend. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we need to find the value of n such that 6n x n is less than or equal to 637.
Step 5: The next step is finding 6n x n ≤ 637. Let us consider n as 9, now 69 x 9 = 621.
Step 6: Subtract 621 from 637, the difference is 16, and the quotient is 39.
Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 1600.
Step 8: Now we need to find the new divisor that is 391 because 391 x 4 = 1564.
Step 9: Subtracting 1564 from 1600, we get the result 36.
Step 10: Now the quotient is 39.1
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values, continue until the remainder is zero.
So the square root of √1537 is approximately 39.197.
Square Root of 1537 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 1537 using the approximation method.
Step 1: Now we have to find the closest perfect squares of √1537.
The smallest perfect square less than 1537 is 1521, and the largest perfect square greater than 1537 is 1600. √1537 falls somewhere between 39 and 40.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (1537 - 1521) ÷ (1600 - 1521) = 16 ÷ 79 ≈ 0.2025
Using the formula, we identify the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 39 + 0.2025 ≈ 39.2025. So the square root of 1537 is approximately 39.2.
Common Mistakes and How to Avoid Them in the Square Root of 1537
Students often make mistakes while finding the square root, such as forgetting about the negative square root, skipping long division methods, etc. 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 principal 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 second 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 1537 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √1537?
The area of the square is approximately 1537 square units.
Explanation
The area of the square = side².
The side length is given as √1537.
Area of the square = side² = √1537 x √1537 ≈ 39.197 x 39.197 ≈ 1537.
Therefore, the area of the square box is approximately 1537 square units.
Problem 2
A square-shaped building measuring 1537 square feet is built; if each of the sides is √1537, what will be the square feet of half of the building?
Approximately 768.5 square feet.
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 1537 by 2, we get approximately 768.5.
So half of the building measures approximately 768.5 square feet.
Problem 3
Calculate √1537 x 5.
Approximately 195.985.
Explanation
The first step is to find the square root of 1537, which is approximately 39.197.
The second step is to multiply 39.197 by 5.
So, 39.197 x 5 ≈ 195.985.
Problem 4
What will be the square root of (1500 + 37)?
The square root is approximately 39.197.
Explanation
To find the square root, we need to find the sum of (1500 + 37). 1500 + 37 = 1537, and then √1537 ≈ 39.197.
Therefore, the square root of (1500 + 37) is approximately ±39.197.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √1537 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 154.394 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√1537 + 38) ≈ 2 × (39.197 + 38) ≈ 2 × 77.197 ≈ 154.394 units.
FAQ on Square Root of 1537
1.What is √1537 in its simplest form?
2.Mention the factors of 1537.
3.Calculate the square of 1537.
4.Is 1537 a prime number?
5.1537 is divisible by?
6.How does learning Algebra help students in Australia make better decisions in daily life?
7.How can cultural or local activities in Australia support learning Algebra topics such as Square Root of 1537?
8.How do technology and digital tools in Australia support learning Algebra and Square Root of 1537?
9.Does learning Algebra support future career opportunities for students in Australia?
Important Glossaries for the Square Root of 1537
- Square root: A square root is the inverse of a square. Example: 4² = 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.
- Long division method: A method used to find the square root of a non-perfect square number by dividing the number into pairs of digits, starting from the decimal point.
- Approximation method: A method for estimating the square root of a number by finding nearby perfect squares and interpolating between them.
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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.
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