Square Root of 3591

The square root is the inverse of the square of a number. 3591 is not a perfect square. The square root of 3591 is expressed in both radical and exponential form. In radical form, it is expressed as √3591, whereas in exponential form, it is expressed as (3591)^(1/2). √3591 = 59.9033, 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, for non-perfect square numbers, methods like the long division method and approximation method are used. Let us now learn about these 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 3591 is broken down into its prime factors.

Step 1: Finding the prime factors of 3591 Breaking it down, we get 3 x 3 x 11 x 109.

Step 2: Now we have found the prime factors of 3591. The second step is to make pairs of those prime factors. Since 3591 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating √3591 using prime factorization is not feasible.

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 3591, we need to group it as 59 and 35.

Step 2: Now we need to find n whose square is less than or equal to 35. We can say n as ‘5’ because 5 x 5 = 25, which is less than 35. Now the quotient is 5 and after subtracting 25 from 35, the remainder is 10.

Step 3: Now let us bring down 91, making the new dividend 1091. Add the old divisor with the same number 5 + 5 to get 10, which will be part of our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 10n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 10n × n ≤ 1091. Let us consider n as 9, so 109 x 9 = 981.

Step 6: Subtract 981 from 1091, the difference is 110, and the quotient is 59.

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, making it 11000.

Step 8: Now we need to find the new divisor, which is 599 because 599 x 1 = 599.

Step 9: Subtracting 599 from 11000, we get the result 10401.

Step 10: Now the quotient is 59.9.

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 √3591 is approximately 59.90.

The approximation method is another method for finding square roots; it is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 3591 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √3591. The smallest perfect square less than 3591 is 3481 (59^2) and the largest perfect square more than 3591 is 3600 (60^2). √3591 falls somewhere between 59 and 60.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (3591 - 3481) / (3600 - 3481) = 110 / 119 ≈ 0.924 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: 59 + 0.924 = 59.924, so the square root of 3591 is approximately 59.924.

• 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, so √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 the positive square root that is more commonly used, known as the principal square root.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors.
   
• Long division method: A method used to calculate the square root of a number by dividing the number into pairs of digits and finding the quotient step by step.