Square Root of 4969

The square root is the inverse of the square of the number. 4969 is not a perfect square. The square root of 4969 is expressed in both radical and exponential form. In the radical form, it is expressed as √4969, whereas (4969)^(1/2) in the exponential form. √4969 ≈ 70.477, 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.

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

The product of prime factors is the prime factorization of a number. Now let us look at how 4969 is broken down into its prime factors.

Step 1: Finding the prime factors of 4969

Breaking it down, we get 7 x 7 x 101: 7^2 x 101^1

Step 2: Now we found out the prime factors of 4969. The second step is to make pairs of those prime factors. Since 4969 is not a perfect square, therefore the digits of the number can’t be grouped in pair. Therefore, calculating √4969 using prime factorization is not straightforward.

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 4969, we need to group it as 69 and 49.

Step 2: Now we need to find n whose square is less than or equal to 49. We can say n as ‘7’ because 7 x 7 = 49. Now the quotient is 7 after subtracting 49 - 49, the remainder is 0.

Step 3: Now let us bring down 69, which is the new dividend. Add the old divisor with the same number 7 + 7, we get 14, which will be our new divisor.

Step 4: The new divisor will be 140n, where n is a digit such that 140n x n ≤ 6900.

Step 5: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue until the remainder is zero.

So the square root of √4969 ≈ 70.48

The approximation method is another method for finding the 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 4969 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √4969. The smallest perfect square less than 4969 is 4900, and the largest perfect square greater than 4969 is 5041. √4969 falls somewhere between 70 and 71.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (4969 - 4900) / (5041 - 4900) = 69 / 141 ≈ 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 70 + 0.49 ≈ 70.49.

• 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, that 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, 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.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.

• Long division method: A method used to find the square root of non-perfect squares by using division and subtraction iteratively to approximate the value.