109 LearnersLast updated on May 26th, 2025
Square Root of 171
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 171.
What is the Square Root of 171?
The square root is the inverse of the square of the number. 171 is not a perfect square. The square root of 171 is expressed in both radical and exponential forms. In the radical form, it is expressed as √171, whereas (171)(1/2) in exponential form. √171 ≈ 13.0767, 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 171
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 171 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 171 is broken down into its prime factors.
Step 1: Finding the prime factors of 171 Breaking it down, we get 3 x 3 x 19: 3² x 19
Step 2: Now we found out the prime factors of 171. The second step is to make pairs of those prime factors. Since 171 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 171 using prime factorization is not straightforward.
Square Root of 171 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, we need to group the numbers from right to left. In the case of 171, we need to group it as 71 and 1.
Step 2: Now we need to find n whose square is ≤ 1. We can say n is ‘1’ because 1 x 1 is lesser than or equal to 1. Now the quotient is 1, and after subtracting 1 from 1, the remainder is 0.
Step 3: Now let us bring down 71, 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 20 since we add a digit and guess n such that 2n x n is less than or equal to 71.
Step 5: The next step is finding 2n x n ≤ 71. Let us consider n as 3, now 23 x 3 = 69.
Step 6: Subtract 71 from 69, the difference is 2, and the quotient is 13.
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 200.
Step 8: Now we need to find the new divisor that is 261 because 261 x 7 ≈ 1827.
Step 9: Subtracting 1827 from 2000, we get the result 173.
Step 10: Now the quotient is 13.0.
Step 11: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.
So the square root of √171 ≈ 13.08.
Square Root of 171 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 171 using the approximation method.
Step 1: Now we have to find the closest perfect square to √171. The smallest perfect square less than 171 is 169, and the largest perfect square greater than 171 is 196. √171 falls somewhere between 13 and 14.
Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (171 - 169) / (196 - 169) = 2/27 ≈ 0.074. 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 13 + 0.074 ≈ 13.074.
So the square root of 171 is approximately 13.074.
Common Mistakes and How to Avoid Them in the Square Root of 171
Students do make mistakes while finding the square root, such as forgetting about the negative square root, skipping steps in the long division method, 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 = ±7.071, there is also -7.071 which should not be forgotten.
Square Root of 171 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √171?
The area of the square is 171 square units.
Explanation
The area of the square = side².
The side length is given as √171.
Area of the square = side²
= √171 x √171
= 171.
Therefore, the area of the square box is 171 square units.
Problem 2
A square-shaped building measuring 171 square feet is built; if each of the sides is √171, what will be the square feet of half of the building?
85.5 square feet.
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 171 by 2 = 85.5.
So half of the building measures 85.5 square feet.
Problem 3
Calculate √171 x 5.
65.3835
Explanation
The first step is to find the square root of 171, which is approximately 13.0767.
The second step is to multiply 13.0767 with 5.
So 13.0767 x 5 ≈ 65.3835.
Problem 4
What will be the square root of (150 + 21)?
The square root is approximately 13.
Explanation
To find the square root, we need to find the sum of (150 + 21).
150 + 21 = 171, and then √171 ≈ 13.0767.
Therefore, the square root of (150 + 21) is approximately ±13.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √171 units and the width ‘w’ is 40 units.
The perimeter of the rectangle is approximately 106.1534 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√171 + 40)
= 2 × (13.0767 + 40)
≈ 2 × 53.0767
≈ 106.1534 units.
FAQ on Square Root of 171
1.What is √171 in its simplest form?
2.Mention the factors of 171.
3.Calculate the square of 171.
4.Is 171 a prime number?
5.171 is divisible by?
6.How does learning Algebra help students in India make better decisions in daily life?
7.How can cultural or local activities in India support learning Algebra topics such as Square Root of 171?
8.How do technology and digital tools in India support learning Algebra and Square Root of 171?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of 171
- 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.
- Prime factorization: Breaking down a number into its basic prime number components.
- Decimal approximation: The decimal representation of an irrational number for practical calculations.
- Long division method: A technique used to find the square root of non-perfect square numbers by using division and subtraction iterations.
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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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