113 LearnersLast updated on May 26th, 2025
Square Root of 157
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 field of vehicle design, finance, etc. Here, we will discuss the square root of 157
What is the Square Root of 157?
The square root is the inverse of the square of the number. 157 is not a perfect square. The square root of 157 is expressed in both radical and exponential form. In the radical form, it is expressed as √157, whereas (157)(1/2) in the exponential form. √157 = 12.52996, 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 157
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where 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 157 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 157 is broken down into its prime factors
Step 1: Finding the prime factors of 157 157 is a prime number, which means it cannot be further divided into other prime factors. Thus, prime factorization of 157 is 157 itself.
Step 2: Since 157 is not a perfect square, calculating 157 using prime factorization directly is not feasible for finding the square root.
Square Root of 157 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 157, we need to group it as 57 and 1.
Step 2: Now we need to find n whose square is 1. We can say n as ‘1’ because 1×1 is lesser than or equal to 1. Now the quotient is 1; after subtracting 1-1, the remainder is 0.
Step 3: Now let us bring down 57, which is the new dividend. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor; we need to find the value of n.
Step 5: The next step is finding 2n × n ≤ 57. Let us consider n as 2, now 2×2×2 = 48.
Step 6: Subtract 57 from 48; the difference is 9, and the quotient is 12.
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 900.
Step 8: Now we need to find the new divisor that is 24 because 244 × 4 = 976.
Step 9: Subtracting 976 from 900, we get the result -76.
Step 10: Now the quotient is 12.5.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.
So the square root of √157 is approximately 12.53
Square Root of 157 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 157 using the approximation method.
Step 1: Now we have to find the closest perfect square of √157. The smallest perfect square of 157 is 144, and the largest perfect square of 157 is 169. √157 falls somewhere between 12 and 13.
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 (157 - 144) / (169 - 144) = 0.52 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 12 + 0.52 = 12.52.
So the square root of 157 is approximately 12.53
Common Mistakes and How to Avoid Them in the Square Root of 157
Students do make mistakes while finding the square root, like 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 required one.
For example: √50 = 7.07, there is also -7.07 which should not be forgotten.
Square Root of 157 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √157?
The area of the square is 157 square units.
Explanation
The area of the square = side^2.
The side length is given as √157.
Area of the square = side^2
= √157 × √157
= 157.
Therefore, the area of the square box is 157 square units.
Problem 2
If a square-shaped building measuring 157 square feet is built, and each of the sides is √157, what will be the square feet of half of the building?
78.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 157 by 2 = we get 78.5
So half of the building measures 78.5 square feet.
Problem 3
Calculate √157 × 5.
Approximately 62.65
Explanation
The first step is to find the square root of 157, which is approximately 12.53.
The second step is to multiply 12.53 with 5.
So 12.53 × 5 = approximately 62.65
Problem 4
What will be the square root of (144 + 13)?
The square root is 13
Explanation
To find the square root, we need to find the sum of (144 + 13).
144 + 13 = 157, and then √157 is approximately 12.53.
Therefore, the square root of (144 + 13) is ±12.53
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √157 units and the width ‘w’ is 20 units.
We find the perimeter of the rectangle to be approximately 65.06 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√157 + 20)
= 2 × (12.53 + 20)
= 2 × 32.53
= approximately 65.06 units.
FAQ on Square Root of 157
1.What is √157 in its simplest form?
2.Mention the factors of 157.
3.Calculate the square of 157.
4.Is 157 a prime number?
5.157 is divisible by?
6.How does learning Algebra help students in Canada make better decisions in daily life?
7.How can cultural or local activities in Canada support learning Algebra topics such as Square Root of 157?
8.How do technology and digital tools in Canada support learning Algebra and Square Root of 157?
9.Does learning Algebra support future career opportunities for students in Canada?
Important Glossaries for the Square Root of 157
- 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.
- Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
- Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups and performing division iteratively.
- Approximation method: A method used to estimate the square root by comparing it to the nearest perfect squares and interpolating for a more precise value.
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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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