Square Root of 2989

The square root is the inverse of the square of the number. 2989 is not a perfect square. The square root of 2989 is expressed in both radical and exponential form. In the radical form, it is expressed as √2989, whereas (2989)^(1/2) in the exponential form. √2989 ≈ 54.686, 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 2989 is broken down into its prime factors.

Step 1: Finding the prime factors of 2989 Breaking it down, we get 29 x 103: 29^1 x 103^1

Step 2: Now we found out the prime factors of 2989. The second step is to make pairs of those prime factors. Since 2989 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 2989 using prime factorization is impossible.

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 2989, we need to group it as 89 and 29.

Step 2: Now we need to find n whose square is less than or equal to 29. We can say n as ‘5’ because 5 x 5 = 25, which is less than 29. Now the quotient is 5, and after subtracting 25 from 29, the remainder is 4.

Step 3: Now let us bring down 89, which is the new dividend. Add the old divisor with the same number 5 + 5, we get 10, which will be our new divisor.

Step 4: The new divisor will be 10n. We need to find the value of n such that 10n x n is less than or equal to 489.

Step 5: The next step is finding 10n x n ≤ 489. Let us consider n as 4, now 104 x 4 = 416.

Step 6: Subtract 416 from 489; the difference is 73, and the quotient is 54.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 7300.

Step 8: Now we need to find the new divisor that is 109 because 1099 x 9 = 9891, which is less than 7300.

Step 9: Subtracting 9891 from 7300; we get the result 1591.

Step 10: Now the quotient is 54.6

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero. So the square root of √2989 is approximately 54.68.

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 2989 using the approximation method.

Step 1: Now we have to find the closest perfect square of √2989. The smallest perfect square less than 2989 is 2809, and the largest perfect square greater than 2989 is 3025. √2989 falls somewhere between 53 and 55.

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 (2989 - 2809) / (3025 - 2809) = 180 / 216 = 0.8333 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 53 + 0.8333 ≈ 53.83, so the square root of 2989 is approximately 54.68.

• 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 a 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: It is the process of expressing a number as the product of its prime factors. For example, 2989 = 29 x 103.

• Long division method: A method used to find the square root of non-perfect squares by dividing the number into pairs of digits and using long division.