Square Root of 67.72
2026-02-28 19:18 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-67.72

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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 the field of vehicle design, finance, etc. Here, we will discuss the square root of 67.72

What is the Square Root of 67.72?

The square root is the inverse of the square of a number. 67.72 is not a perfect square. The square root of 67.72 is expressed in both radical and exponential form.

In the radical form, it is expressed as √67.72, whereas (67.72)1/2 in the exponential form. √67.72 ≈ 8.228, 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 67.72

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:

  1. Prime factorization method
  2. Long division method
  3. Approximation method

Square Root of 67.72 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. However, since 67.72 is not an integer, prime factorization is not applicable in the traditional sense.

For non-perfect squares like 67.72, we focus on other methods such as the long division or approximation method.

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Square Root of 67.72 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 67.72, we consider two decimal places for precision.

Step 2: Identify the largest number whose square is less than or equal to 67. The number is 8 because 8 x 8 = 64. Now, the quotient is 8 and the remainder is 3.

Step 3: Bring down 72 to make the new dividend 372. Double the quotient and bring down the next pair of digits. We get 16_ as the new divisor.

Step 4: Find the largest digit (n) such that (16n) * n ≤ 372. Let n be 2; thus, (162) x 2 = 324.

Step 5: Subtract 324 from 372 to get a remainder of 48. In the quotient, we have 8.2.

Step 6: Since we want more decimal precision, bring down pairs of zeros to continue the process.

Step 7: Continue the steps until desired precision is reached. The square root of 67.72 is approximately 8.228.

Square Root of 67.72 by Approximation Method

The approximation method is another method for finding square roots. It is an easy method to estimate the square root of a given number. Let's learn how to find the square root of 67.72 using the approximation method.

Step 1: Identify the closest perfect squares around 67.72. The closest perfect squares are 64 and 81, where √64 = 8 and √81 = 9.

Step 2: 67.72 is closer to 64 than it is to 81. We can estimate that √67.72 is slightly more than 8.

Step 3: Use linear approximation or interpolation to refine the estimate. For example: (67.72 - 64) / (81 - 64) = 0.219, so an additional 0.219 x (9 - 8) = 0.219 should be added to 8.

Therefore, √67.72 ≈ 8.219, but refining this further with more precise calculations gives √67.72 ≈ 8.228.

Common Mistakes and How to Avoid Them in Finding the Square Root of 67.72

Students often make mistakes while finding square roots, such as forgetting about the negative square root, skipping long division method steps, etc. Now let us look at a few of those mistakes in detail.

Problem 1

Can you help Max find the area of a square box if its side length is given as √67?

Okay, lets begin

The area of the square is approximately 67 square units.

Explanation

The area of the square = side².

The side length is given as √67.

Area of the square = side² = (√67) x (√67) = 67.

Therefore, the area of the square box is 67 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 67.72 square feet is built; if each of the sides is √67.72, what will be the square feet of half of the building?

Okay, lets begin

33.86 square feet

Explanation

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

Dividing 67.72 by 2, we get 33.86.

So half of the building measures 33.86 square feet.

Well explained 👍

Problem 3

Calculate √67.72 x 5.

Okay, lets begin

Approximately 41.14

Explanation

The first step is to find the square root of 67.72, which is approximately 8.228.

The second step is to multiply 8.228 with 5. So, 8.228 x 5 ≈ 41.14.

Well explained 👍

Problem 4

What will be the square root of (67 + 1)?

Okay, lets begin

Approximately 8.25

Explanation

To find the square root, we need to find the sum of (67 + 1), which is 68. √68 ≈ 8.25.

Therefore, the square root of (67 + 1) is approximately ±8.25.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √67.72 units and the width ‘w’ is 38 units.

Okay, lets begin

The perimeter of the rectangle is approximately 92.456 units.

Explanation

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

Perimeter = 2 × (√67.72 + 38)

= 2 × (8.228 + 38)

= 2 × 46.228 = 92.456 units.

Well explained 👍

FAQ on Square Root of 67.72

1.What is √67.72 in its simplest form?

Since 67.72 is not a perfect square, √67.72 is not simplified further in terms of integers.

It remains as √67.72 ≈ 8.228.

2.What are the factors of 67.72?

67.72 is not an integer, so it doesn't have factors in the traditional sense like whole numbers do. However, it can be expressed as a product of its prime factors when considered as a decimal approximation.

3.Calculate the square of 67.72.

We get the square of 67.72 by multiplying the number by itself, that is 67.72 x 67.72 ≈ 4586.7584.

4.Is 67.72 a prime number?

67.72 is not a prime number, as it is not an integer and cannot be classified traditionally as prime or composite.

5.67.72 is divisible by?

67.72 is not an integer and thus not typically divided like whole numbers. However, as a decimal, it can be divided by 1 or any factor of its whole-number approximation.

Important Glossaries for the Square Root of 67.72

  • Square root: A square root is the inverse of a square. For 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, we often consider only the positive square root, known as the principal square root.
  • Decimals: If a number has a whole number and a fraction in a single number, it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.
  • Approximation: Approximation is the process of finding a value that is close to the actual value. For example: The square root of 67.72 is approximately 8.228.

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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.