Square Root of 3969
2026-02-28 18:00 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-3969

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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 3969.

What is the Square Root of 3969?

The square root is the inverse of the square of a number. 3969 is a perfect square. The square root of 3969 is expressed in both radical and exponential forms. In the radical form, it is expressed as √3969, whereas (3969)^(1/2) is the exponential form. √3969 = 63, which is a rational number because it can be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 3969

The prime factorization method is particularly useful for perfect square numbers. For non-perfect square numbers, 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 3969 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 3969 is broken down into its prime factors.

Step 1: Finding the prime factors of 3969 Breaking it down, we get 3 × 3 × 3 × 3 × 7 × 7: 3^4 × 7^2

Step 2: Now we found out the prime factors of 3969. The second step is to make pairs of those prime factors. Since 3969 is a perfect square, we can group the digits in pairs. Step 3: Taking one number from each pair and multiplying, we get: 3 × 3 × 7 = 63

Therefore, the square root of 3969 is 63.

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Square Root of 3969 by Long Division Method

The long division method is particularly used for non-perfect square numbers, but it can also be used for perfect squares. 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 3969, we need to group it as 69 and 39.

Step 2: Now we need to find n whose square is ≤ 39. We can say n is ‘6’ because 6 × 6 = 36, which is lesser than or equal to 39. Now the quotient is 6, and after subtracting 39 - 36, the remainder is 3.

Step 3: Now let us bring down 69, which is the new dividend. Add the old divisor with the same number 6 + 6, we get 12, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 12n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 12n × n ≤ 369. Let us consider n as 3, now 12 × 3 × 3 = 369.

Step 6: Subtract 369 from 369, and the difference is 0, and the quotient is 63.

Since the remainder is zero, the square root of 3969 is 63.

Square Root of 3969 by Approximation Method

The approximation method is another way for finding square roots, especially useful for non-perfect squares, but can also verify perfect squares. Now let us learn how to find the square root of 3969 using the approximation method.

Step 1: Now we have to find the closest perfect square of √3969. The closest perfect squares are 3600 (60^2) and 4096 (64^2). √3969 falls between 60 and 64, but since 3969 is a perfect square, we know the exact square root is 63.

Step 2: Since we've identified 3969 as a perfect square, there is no need for further approximation. We know that √3969 = 63.

Common Mistakes and How to Avoid Them in the Square Root of 3969

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 √3249?

Okay, lets begin

The area of the square is 3249 square units.

Explanation

The area of the square = side².

The side length is given as √3249.

Area of the square = side² = √3249 × √3249 = 57 × 57 = 3249.

Therefore, the area of the square box is 3249 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 3969 square feet is built; if each of the sides is √3969, what will be the square feet of half of the building?

Okay, lets begin

1984.5 square feet

Explanation

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

Dividing 3969 by 2 = 1984.5.

So half of the building measures 1984.5 square feet.

Well explained 👍

Problem 3

Calculate √3969 × 5.

Okay, lets begin

315

Explanation

The first step is to find the square root of 3969, which is 63.

The second step is to multiply 63 with 5.

So 63 × 5 = 315.

Well explained 👍

Problem 4

What will be the square root of (3249 + 720)?

Okay, lets begin

The square root is 69.

Explanation

To find the square root, we need to find the sum of (3249 + 720).

3249 + 720 = 3969, and then √3969 = 63.

Therefore, the square root of (3249 + 720) is ±63.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √3249 units and the width ‘w’ is 38 units.

Okay, lets begin

We find the perimeter of the rectangle as 190 units.

Explanation

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

Perimeter = 2 × (√3249 + 38)

= 2 × (57 + 38)

= 2 × 95

= 190 units.

Well explained 👍

FAQ on Square Root of 3969

1.What is √3969 in its simplest form?

The prime factorization of 3969 is 3 × 3 × 3 × 3 × 7 × 7, so the simplest form of √3969 = √(3^4 × 7^2) = 63.

2.Mention the factors of 3969.

Factors of 3969 are 1, 3, 9, 21, 27, 63, 189, 441, 1323, 3969.

3.Calculate the square of 3969.

We get the square of 3969 by multiplying the number by itself, that is 3969 × 3969 = 15,747,561.

4.Is 3969 a prime number?

3969 is not a prime number, as it has more than two factors.

5.3969 is divisible by?

3969 has many factors; those are 1, 3, 9, 21, 27, 63, 189, 441, 1323, 3969.

Important Glossaries for the Square Root of 3969

  • Square root: A square root is the inverse of a square. Example: 8² = 64, and the inverse of the square is the square root, that is √64 = 8.
     
  • Rational number: A rational number is a number that can be written in the form of p/q, where q is not equal to zero and p and q are integers.
     
  • Perfect square: A perfect square is a number that has an integer as its square root. For example, 64 is a perfect square because its square root is 8.
     
  • Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors.
     
  • Long division method: A method used to find the square root of a number by dividing it into smaller, more manageable parts.

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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.