Square Root of 217
2026-02-28 21:36 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-217

304 Learners

Last updated on August 5, 2025

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 such as vehicle design, finance, etc. Here, we will discuss the square root of 217.

What is the Square Root of 217?

The square root is the inverse of the square of the number. 217 is not a perfect square. The square root of 217 is expressed in both radical and exponential form. In the radical form, it is expressed as √217, whereas (217)^(1/2) in the exponential form. √217 ≈ 14.73092, 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 217

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 217 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 217 is broken down into its prime factors:

Step 1: Finding the prime factors of 217 217 is a prime number, which means it only has two factors: 1 and 217.

Therefore, using prime factorization to simplify √217 is not possible.

Explore Our Programs

Square Root of 217 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 217, we group it as 17 and 2.

Step 2: Now we need to find n whose square is less than or equal to 2. We can say n is ‘1’ because 1 × 1 is less than or equal to 2. The quotient is 1, and after subtracting, we have a remainder of 1.

Step 3: Bring down 17, making the new dividend 117. Add the old divisor with the same number, 1 + 1, to get 2, which will be our new divisor.

Step 4: We need to find the largest digit n such that 2n × n ≤ 117. Let n be 4, then 2 × 4 × 4 = 64.

Step 5: Subtract 64 from 117, the difference is 53, and the quotient is 14.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point, allowing us to bring down pairs of zeros.

Step 7: Continue the long division process until you reach the desired decimal places, resulting in √217 ≈ 14.73.

Square Root of 217 by Approximation Method

The approximation method is another method for finding square roots. It is an easy way to find the square root of a given number. Now let us learn how to find the square root of 217 using the approximation method:

Step 1: Find the closest perfect square numbers to 217. The smallest perfect square less than 217 is 196 (14²), and the largest perfect square greater than 217 is 225 (15²). √217 falls between 14 and 15.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Largest perfect square - smallest perfect square). Using the formula (217 - 196) ÷ (225 - 196) = 21 ÷ 29 ≈ 0.724. Adding this to the smaller perfect square root, we get 14 + 0.724 ≈ 14.724.

Common Mistakes and How to Avoid Them in the Square Root of 217

Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. Let us look at a few of these mistakes in detail.

Download Worksheets

Problem 1

Can you help Max find the area of a square box if its side length is given as √217?

Okay, lets begin

The area of the square is approximately 217 square units.

Explanation

The area of the square = side².

The side length is given as √217.

Area of the square = side² = √217 × √217 = 217.

Therefore, the area of the square box is approximately 217 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 217 square feet is built; if each of the sides is √217, what will be the square feet of half of the building?

Okay, lets begin

108.5 square feet

Explanation

We can divide the given area by 2 since the building is square-shaped.

Dividing 217 by 2 = 108.5.

So half of the building measures 108.5 square feet.

Well explained 👍

Problem 3

Calculate √217 × 5.

Okay, lets begin

Approximately 73.65

Explanation

First, find the square root of 217, which is approximately 14.73.

Then multiply 14.73 by 5. So, 14.73 × 5 ≈ 73.65.

Well explained 👍

Problem 4

What will be the square root of (207 + 10)?

Okay, lets begin

The square root is approximately 14.73.

Explanation

To find the square root, first calculate the sum of (207 + 10).

207 + 10 = 217, then √217 ≈ 14.73.

Therefore, the square root of (207 + 10) is approximately 14.73.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √217 units and the width ‘w’ is 40 units.

Okay, lets begin

The perimeter of the rectangle is approximately 109.46 units.

Explanation

Perimeter of the rectangle = 2 × (length + width).

Perimeter = 2 × (√217 + 40) = 2 × (14.73 + 40) = 2 × 54.73 ≈ 109.46 units.

Well explained 👍

FAQ on Square Root of 217

1.What is √217 in its simplest form?

Since 217 is a prime number, √217 is already in its simplest radical form.

2.Mention the factors of 217.

Factors of 217 are 1 and 217 because 217 is a prime number.

3.Calculate the square of 217.

We get the square of 217 by multiplying the number by itself, that is 217 × 217 = 47,089.

4.Is 217 a prime number?

Yes, 217 is a prime number, as it only has two factors: 1 and 217.

5.217 is divisible by?

217 is divisible only by 1 and 217 as it is a prime number.

Important Glossaries for the Square Root of 217

  • 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 number: A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
  • Decimal approximation: The process of finding a decimal number that is close to an exact value.
  • Long division method: A technique used to find the square root of non-perfect squares by dividing and averaging in successive steps.

What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math

▶

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

: He loves to play the quiz with kids through algebra to make kids love it.