Square Root of 577
2026-02-28 09:58 Diff
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Last updated on September 30, 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 like vehicle design, finance, etc. Here, we will discuss the square root of 577.

What is the Square Root of 577?

The square root is the inverse of the square of the number. 577 is not a perfect square. The square root of 577 is expressed in both radical and exponential form. In radical form, it is expressed as √577, whereas in exponential form it is expressed as (577)^(1/2). √577 ≈ 24.0208, 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 577

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, the long division method and the approximation method are used. Let us now learn the following methods:

  • Prime factorization method
     
  • Long division method
     
  • Approximation method

Square Root of 577 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 577 is broken down into its prime factors:

Step 1: Finding the prime factors of 577 577 is a prime number, meaning it cannot be broken down further into smaller prime factors.

Therefore, prime factorization is not applicable here.

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Square Root of 577 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 digits of 577 from right to left as 77 and 5.

Step 2: Now we need to find n whose square is closest to 5. We can say n as ‘2’ because 2 × 2 = 4 is less than or equal to 5. Now the quotient is 2, and after subtracting 4 from 5, the remainder is 1.

Step 3: Bring down 77, making the new dividend 177. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: Find n such that 4n × n ≤ 177. Let us consider n as 4. Now, 44 × 4 = 176.

Step 5: Subtract 176 from 177, the difference is 1, and the quotient is 24.

Step 6: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend, making it 100.

Step 7: Find the new divisor 480 because 480 × 0 = 0.

Step 8: Subtract 0 from 100, and continue the process until you achieve the desired precision. The square root of √577 is approximately 24.0208.

Square Root of 577 by Approximation Method

The approximation method is another way to find 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 577 using the approximation method.

Step 1: Find the closest perfect squares around 577. The smallest perfect square before 577 is 576, and the next perfect square is 625. √577 falls between 24 and 25.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (577 - 576) / (625 - 576) = 0.02

Step 3: Add the value obtained to the smaller square root: 24 + 0.02 = 24.02, so the square root of 577 is approximately 24.02.

Common Mistakes and How to Avoid Them in Finding the Square Root of 577

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

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Problem 1

Can you help Max find the area of a square box if its side length is given as √577?

Okay, lets begin

The area of the square is 577 square units.

Explanation

The area of a square = side².

The side length is given as √577.

Area of the square = √577 × √577 = 577.

Therefore, the area of the square box is 577 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 577 square feet is built; if each of the sides is √577, what will be the square feet of half of the building?

Okay, lets begin

288.5 square feet

Explanation

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

Dividing 577 by 2 gives 288.5.

So half of the building measures 288.5 square feet.

Well explained 👍

Problem 3

Calculate √577 × 5.

Okay, lets begin

120.104

Explanation

First, find the square root of 577, which is approximately 24.0208.

Multiply it by 5. So, 24.0208 × 5 ≈ 120.104.

Well explained 👍

Problem 4

What will be the square root of (576 + 1)?

Okay, lets begin

The square root is approximately 24.0208.

Explanation

To find the square root, sum (576 + 1) = 577, then √577 ≈ 24.0208.

Therefore, the square root of (576 + 1) is approximately ±24.0208.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √577 units and the width ‘w’ is 23 units.

Okay, lets begin

The perimeter of the rectangle is approximately 94.0416 units.

Explanation

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

Perimeter = 2 × (√577 + 23) ≈ 2 × (24.0208 + 23) = 2 × 47.0208 ≈ 94.0416 units.

Well explained 👍

FAQ on Square Root of 577

1.What is √577 in its simplest form?

Since 577 is a prime number, its simplest radical form is √577.

2.Mention the factors of 577.

The factors of 577 are 1 and 577 as it is a prime number.

3.Calculate the square of 577.

We get the square of 577 by multiplying the number by itself, that is 577 × 577 = 333,129.

4.Is 577 a prime number?

Yes, 577 is a prime number as it has only two factors: 1 and 577.

5.577 is divisible by?

577 is only divisible by 1 and 577 since it is a prime number.

Important Glossaries for the Square Root of 577

  • Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: √16 = 4.
  • Irrational number: An irrational number cannot be written as a simple fraction, i.e., in the form of p/q, where q is not equal to zero and p and q are integers.
  • Prime number: A number greater than 1 that has no positive divisors other than 1 and itself. Example: 577.
  • Long division method: A technique used to find the approximate square root of a number, especially for non-perfect squares.
  • Decimal approximation: The process of finding an approximate decimal value for a square root that is not a perfect square.

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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.