Square Root of 272
2026-02-28 23:13 Diff
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Last updated on August 5, 2025

If a number is multiplied by itself, the result is a square. The inverse operation of finding the square is calculating the square root. Square roots are applied in various fields such as engineering, physics, and statistics. Here, we will discuss the square root of 272.

What is the Square Root of 272?

The square root is the inverse of squaring a number. 272 is not a perfect square. The square root of 272 can be expressed in both radical and exponential forms. In radical form, it is expressed as √272, whereas in exponential form it is expressed as (272)^(1/2). √272 ≈ 16.49242, which is an irrational number because it cannot be expressed as a ratio of two integers.

Finding the Square Root of 272

The prime factorization method is typically used for perfect square numbers. However, for non-perfect square numbers like 272, the long division method and approximation method are commonly used. Let's explore the following methods:

  • Prime factorization method 
  • Long division method 
  • Approximation method

Square Root of 272 by Prime Factorization Method

The prime factorization of a number is the product of its prime factors. Let's break down 272 into its prime factors:

Step 1: Finding the prime factors of 272 Breaking it down, we get 2 × 2 × 2 × 2 × 17: 2^4 × 17

Step 2: Now that we have found the prime factors of 272, the next step is to make pairs of those prime factors. Since 272 is not a perfect square, the digits cannot be grouped into pairs that form a perfect square, and thus calculating the square root using prime factorization is not straightforward.

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Square Root of 272 by Long Division Method

The long division method is particularly useful for non-perfect square numbers. In this method, we find the closest perfect square number to the given number. Let's learn how to find the square root using the long division method, step by step:

Step 1: Start by grouping the digits in pairs from right to left. For 272, we group it as 72 and 2.

Step 2: Find a number n whose square is less than or equal to 2. Here, n is 1 because 1 × 1 ≤ 2. The quotient is 1, and the remainder is 1 after subtracting 1 from 2.

Step 3: Bring down 72 to make the new dividend 172. Double the previous divisor and append a digit (x) to form a new divisor, which is 2x.

Step 4: Find a digit x such that 2x × x ≤ 172. Let x = 6, then 26 × 6 = 156.

Step 5: Subtract 156 from 172, giving a remainder of 16. The quotient is 16.

Step 6: Bring down two zeros to make the new dividend 1600.

Step 7: The new divisor is 320 (since 32 × 5 = 1600), and the next digit of the quotient is 5.

Step 8: Continue this process until you achieve the desired precision. The square root of √272 is approximately 16.49.

Square Root of 272 by Approximation Method

The approximation method is a straightforward way to find square roots. Here's how to find the square root of 272 using approximation:

Step 1: Identify the closest perfect squares to 272. The smallest perfect square less than 272 is 256, and the largest perfect square greater than 272 is 289. Thus, √272 is between 16 and 17.

Step 2: Use the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) Applying the formula: (272 - 256) / (289 - 256) = 16 / 33 ≈ 0.48

Step 3: Add this decimal to the lower bound: 16 + 0.48 = 16.48 Thus, the approximate square root of 272 is 16.48.

Common Mistakes and How to Avoid Them in the Square Root of 272

Students often make mistakes when finding square roots, such as forgetting about the negative square root or skipping steps in the long division method. Let's look at some common mistakes and how to avoid them.

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Problem 1

Can you help Max find the area of a square box if its side length is given as √272?

Okay, lets begin

The area of the square is 272 square units.

Explanation

The area of the square = side².

The side length is given as √272.

Area of the square = (√272)² = 272.

Therefore, the area of the square box is 272 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 272 square feet is built; if each of the sides is √272, what will be the square feet of half of the building?

Okay, lets begin

136 square feet

Explanation

To find the area of half the building, divide the total area by 2.

Dividing 272 by 2 = 136.

So half of the building measures 136 square feet.

Well explained 👍

Problem 3

Calculate √272 × 5.

Okay, lets begin

Approximately 82.46

Explanation

First, find the square root of 272, which is approximately 16.49.

Then multiply 16.49 by 5.

So, 16.49 × 5 ≈ 82.46.

Well explained 👍

Problem 4

What will be the square root of (256 + 16)?

Okay, lets begin

The square root is 17.

Explanation

First, find the sum of (256 + 16). 256 + 16 = 272, and then √272 = 16.49.

Therefore, the square root of (256 + 16) is approximately ±16.49.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √272 units and the width ‘w’ is 38 units.

Okay, lets begin

The perimeter of the rectangle is approximately 109.98 units.

Explanation

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

Perimeter = 2 × (√272 + 38).

Perimeter ≈ 2 × (16.49 + 38) = 2 × 54.49 ≈ 109.98 units.

Well explained 👍

FAQ on Square Root of 272

1.What is √272 in its simplest form?

The prime factorization of 272 is 2 × 2 × 2 × 2 × 17, so the simplest form of √272 = √(2^4 × 17) = 4√17.

2.Mention the factors of 272.

Factors of 272 are 1, 2, 4, 8, 16, 17, 34, 68, 136, and 272.

3.Calculate the square of 272.

The square of 272 is calculated by multiplying the number by itself: 272 × 272 = 73984.

4.Is 272 a prime number?

5.272 is divisible by?

272 has several divisors: 1, 2, 4, 8, 16, 17, 34, 68, 136, and 272.

Important Glossaries for the Square Root of 272

  • Square Root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: The square root of 16 is 4 because 4 × 4 = 16.
  • Irrational Number: An irrational number is a number that cannot be expressed as a simple fraction or ratio of two integers. Example: √2 is an irrational number.
  • Principal Square Root: The principal square root is the non-negative square root of a number. For example, the principal square root of 9 is 3.
  • Prime Factorization: Prime factorization is expressing a number as the product of its prime numbers. Example: The prime factorization of 28 is 2 × 2 × 7.
  • Approximation Method: The approximation method helps estimate the value of square roots for numbers that are not perfect squares, providing a close decimal 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

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