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.

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

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.

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.

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.

• 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.