Square Root of 772
2026-02-28 19:17 Diff
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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 fields such as vehicle design, finance, etc. Here, we will discuss the square root of 772.

What is the Square Root of 772?

The square root is the inverse of the square of a number. 772 is not a perfect square. The square root of 772 is expressed in both radical and exponential form. In the radical form, it is expressed as √772, whereas in exponential form, it is (772)^(1/2). √772 ≈ 27.78489, 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 772

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 772 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 772 is broken down into its prime factors.

Step 1: Finding the prime factors of 772 Breaking it down, we get 2 × 2 × 193: 2^2 × 193

Step 2: Since 772 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 772 using prime factorization alone cannot provide an exact square root.

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Square Root of 772 by Long Division Method

The long division method is particularly used for non-perfect square numbers. Let's learn how to find the square root using the long division method, step by step.

Step 1: Group the digits of 772 from right to left. In this case, we have one group of two digits (72) and one group of one digit (7).

Step 2: Find the largest number whose square is less than or equal to 7. This number is 2, as 2^2 = 4. Subtract 4 from 7, leaving a remainder of 3.

Step 3: Bring down the next pair of digits (72) to get 372.

Step 4: Double the divisor (2) and find the next digit of the quotient such that (20 + n) × n is less than or equal to 372. The number is 7, as 27 × 7 = 189.

Step 5: Subtract 189 from 372 to get 183. Bring down two zeros to make it 18300.

Step 6: Repeat the process to find the next digit in the decimal places. Continue this process to find the square root to the desired precision.

Thus, √772 ≈ 27.78.

Square Root of 772 by Approximation Method

The approximation method is another method for finding square roots. Now let us learn how to approximate the square root of 772.

Step 1: Identify the perfect squares closest to 772.

The smallest perfect square less than 772 is 729, and the largest perfect square greater than 772 is 784. So, √772 falls between 27 and 28.

Step 2: Apply the formula: (Given number - smallest perfect square) / (largest perfect square - smallest perfect square).

Using this formula: (772 - 729) / (784 - 729) = 43 / 55 ≈ 0.7818. Adding the approximate decimal to the lower perfect square root gives 27 + 0.7818 ≈ 27.78.

Common Mistakes and How to Avoid Them in the Square Root of 772

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'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 √772?

Okay, lets begin

The area of the square is approximately 772 square units.

Explanation

The area of the square = side^2.

The side length is given as √772.

Area of the square = (√772)² = 772.

Therefore, the area of the square box is approximately 772 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 772 square feet is built; if each of the sides is √772, what will be the square feet of half of the building?

Okay, lets begin

386 square feet

Explanation

Divide the given area by 2 since the building is square-shaped.

Dividing 772 by 2 gives 386.

So half of the building measures 386 square feet.

Well explained 👍

Problem 3

Calculate √772 × 5.

Okay, lets begin

Approximately 138.92

Explanation

The first step is to find the square root of 772, which is approximately 27.78.

Multiply 27.78 by 5.

So, 27.78 × 5 ≈ 138.92.

Well explained 👍

Problem 4

What will be the square root of (772 + 28)?

Okay, lets begin

The square root is 28.

Explanation

To find the square root, calculate the sum of (772 + 28). 772 + 28 = 800, and then √800 = ±28.284271 (approximately ±28).

Therefore, the approximate square root of (772 + 28) is ±28.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √772 units and the width ‘w’ is 50 units.

Okay, lets begin

The perimeter of the rectangle is approximately 155.56 units.

Explanation

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

Perimeter = 2 × (√772 + 50) ≈ 2 × (27.78 + 50) = 2 × 77.78 ≈ 155.56 units.

Well explained 👍

FAQ on Square Root of 772

1.What is √772 in its simplest form?

The prime factorization of 772 is 2 × 2 × 193, so the simplest form of √772 is √(2 × 2 × 193).

2.Mention the factors of 772.

Factors of 772 are 1, 2, 4, 193, 386, and 772.

3.Calculate the square of 772.

The square of 772 is 772 × 772 = 596,784.

4.Is 772 a prime number?

5.772 is divisible by?

772 is divisible by 1, 2, 4, 193, 386, and 772.

Important Glossaries for the Square Root of 772

  • 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, which 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, the positive square root is often used in real-world applications. This is known as the principal square root.
  • Prime factorization: Prime factorization is the process of expressing a number as a product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.
  • Long division method: The long division method is a step-by-step process used to find the square root of a number, especially useful for non-perfect squares.

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