Square Root of 650
2026-02-28 11:43 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 the field of vehicle design, finance, etc. Here, we will discuss the square root of 650.

What is the Square Root of 650?

The square root is the inverse of the square of a number. 650 is not a perfect square. The square root of 650 is expressed in both radical and exponential form. In the radical form, it is expressed as √650, whereas (650)^(1/2) in the exponential form. √650 ≈ 25.4951, 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 650

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where 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 650 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 650 is broken down into its prime factors.

Step 1: Finding the prime factors of 650 Breaking it down, we get 2 x 5 x 5 x 13: 2^1 x 5^2 x 13^1

Step 2: Now we found out the prime factors of 650. The second step is to make pairs of those prime factors. Since 650 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 650 using prime factorization is impossible.

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Square Root of 650 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 650, we need to group it as 50 and 6.

Step 2: Now we need to find n whose square is less than or equal to 6. We can say n as ‘2’ because 2 x 2 = 4 is lesser than or equal to 6. Now the quotient is 2 after subtracting 6 - 4 the remainder is 2.

Step 3: Now let us bring down 50 which is the new dividend. Add the old divisor with the same number 2 + 2, we get 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find the value of n such that 4n x n ≤ 250. Let us consider n as 5, now 45 x 5 = 225.

Step 5: Subtract 250 from 225; the difference is 25, and the quotient is 25.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 2500.

Step 7: Now we need to find the new divisor. We find it by 510 x 5 = 2550, which is too large, so we try 509 x 4 = 2036.

Step 8: Subtracting 2036 from 2500 gives 464.

Step 9: Now the quotient is 25.4. Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal values continue till the remainder is zero.

So the square root of √650 is approximately 25.495.

Square Root of 650 by Approximation Method

The approximation method is another method for finding 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 650 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √650. The smallest perfect square less than 650 is 625 and the largest perfect square greater than 650 is 676. √650 falls somewhere between 25 and 26.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (650 - 625) / (676 - 625) = 25/51 ≈ 0.49 Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number which is 25 + 0.49 = 25.49, so the square root of 650 is approximately 25.49.

Common Mistakes and How to Avoid Them in the Square Root of 650

Students do make mistakes while finding the square root, like forgetting about the negative square root, skipping long division methods, etc. Now let us look at a few of those mistakes that students tend to make 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 √650?

Okay, lets begin

The area of the square is approximately 650 square units.

Explanation

The area of the square = side^2.

The side length is given as √650.

Area of the square = (√650) x (√650) = 650.

Therefore, the area of the square box is approximately 650 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 650 square feet is built; if each of the sides is √650, what will be the square feet of half of the building?

Okay, lets begin

325 square feet

Explanation

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

Dividing 650 by 2 = we get 325.

So half of the building measures 325 square feet.

Well explained 👍

Problem 3

Calculate √650 x 5.

Okay, lets begin

Approximately 127.475

Explanation

The first step is to find the square root of 650 which is approximately 25.495, the second step is to multiply 25.495 with 5.

So 25.495 x 5 ≈ 127.475.

Well explained 👍

Problem 4

What will be the square root of (650 + 25)?

Okay, lets begin

The square root is approximately 26.

Explanation

To find the square root, we need to find the sum of (650 + 25).

650 + 25 = 675, and then √675 ≈ 25.98.

Therefore, the square root of (650 + 25) is approximately 26.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √650 units and the width ‘w’ is 38 units.

Okay, lets begin

We find the perimeter of the rectangle to be approximately 126.99 units.

Explanation

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

Perimeter = 2 × (√650 + 38)

= 2 × (25.495 + 38)

= 2 × 63.495

≈ 126.99 units.

Well explained 👍

FAQ on Square Root of 650

1.What is √650 in its simplest form?

The prime factorization of 650 is 2 x 5 x 5 x 13, so the simplest form of √650 = √(2 x 5^2 x 13).

2.Mention the factors of 650.

Factors of 650 are 1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, and 650.

3.Calculate the square of 650.

We get the square of 650 by multiplying the number by itself, that is 650 x 650 = 422500.

4.Is 650 a prime number?

5.650 is divisible by?

650 has many factors; those are 1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, and 650.

Important Glossaries for the Square Root of 650

  • Square root: A square root is the inverse of a square. For example: 5^2 = 25, and the inverse of the square is the square root, that is √25 = 5.
     
  • 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, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
     
  • Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 x 3 x 3.
     
  • Long division method: A method used to find the square root of non-perfect squares through repeated division and subtraction, yielding increasingly accurate approximations.

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Jaskaran Singh Saluja

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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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