Square Root of 3589

The square root is the inverse of the square of a number. 3589 is not a perfect square. The square root of 3589 is expressed in both radical and exponential form. In radical form, it is expressed as √3589, whereas (3589)^(1/2) in exponential form. √3589 ≈ 59.87098, 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 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 3589 is broken down into its prime factors:

Step 1: Finding the prime factors of 3589. Breaking it down, we find that 3589 is a prime number and cannot be factored into smaller prime numbers.

Step 2: Since 3589 is a prime number, it cannot be broken into pairs for the prime factorization method. Therefore, calculating 3589 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 3589, we need to group it as 35 and 89.

Step 2: Now, 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. Subtract 25 from 35, and the remainder is 10.

Step 3: Bring down 89 to make it 1089. Add the old divisor with the same number, 5 + 5, to get 10, which will be our new divisor.

Step 4: The new divisor is 10n. We need to find the value of n such that 10n × n ≤ 1089. Let us consider n as 9; now 109 x 9 = 981.

Step 5: Subtract 981 from 1089. The difference is 108, and the quotient is 59.

Step 6: 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 10800.

Step 7: Find the new divisor, which is 598, because 598 x 8 = 4784.

Step 8: Subtract 4784 from 10800, resulting in 6016.

Step 9: Continue this process until you reach the desired decimal precision. The square root of √3589 ≈ 59.87098.

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 3589 using the approximation method.

Step 1: Find the closest perfect squares around √3589. The smallest perfect square less than 3589 is 3481 (59^2), and the largest perfect square greater than 3589 is 3600 (60^2). √3589 falls between 59 and 60.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (3589 - 3481) / (3600 - 3481) ≈ 0.87098. Adding this result to the lower value, 59 + 0.87098 = 59.87098, so the square root of 3589 is approximately 59.87098.

• Square root: A square root is the inverse of a square. Example: 5^2 = 25, and the inverse of the square is the square root, that is, √25 = 5.

• 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. For example, √2 is an irrational number.

• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 3589 is a prime number.

• Decimal approximation: A decimal approximation is an estimate of a number expressed in decimal form, often used for non-perfect squares or other irrational numbers. For example, √3589 ≈ 59.87098.

• Long division method: The long division method is a technique used to find the square root of non-perfect squares through a series of division steps, resulting in a decimal approximation.