Square Root of 3593

The square root is the inverse of the square of the number. 3593 is not a perfect square. The square root of 3593 is expressed in both radical and exponential forms. In the radical form, it is expressed as √3593, whereas (3593)^(1/2) in the exponential form. √3593 ≈ 59.931, 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 typically 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 3593 is broken down into its prime factors.

Step 1: Finding the prime factors of 3593 3593 is a prime number, so it itself is its prime factor.

Step 2: Since 3593 is not a perfect square, calculating its square root 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 3593, we need to group it as 93 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 × 5 = 25 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 93, which is the new dividend. Add the old divisor with the same number 5 + 5 to get 10, which will be our new divisor.

Step 4: We need to find n such that 10n × n ≤ 1093; let us consider n as 9, now 109 × 9 = 981.

Step 5: Subtract 981 from 1093, the difference is 112, and the quotient is 59.

Step 6: Since the dividend is less than the divisor, we add a decimal point to the quotient and add two zeroes to the dividend. The new dividend is 11200.

Step 7: Find the new divisor that is 598, because 598 × 1 = 598.

Step 8: Subtracting 598 from 11200 gives us the remainder 10602.

Step 9: Continue this process until the desired precision is achieved.

The square root of 3593 is approximately 59.931.

The approximation method is another method for finding square roots. It is an easy way to find the square root of a given number. Now let us learn how to find the square root of 3593 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √3593. The smallest perfect square less than 3593 is 3481 (59²) and the largest perfect square greater than 3593 is 3600 (60²). Thus, √3593 falls between 59 and 60.

Step 2: Apply the formula: (Given number - smallest perfect square) / (next perfect square - smallest perfect square) (3593 - 3481) / (3600 - 3481) = 112 / 119 ≈ 0.941 Adding this to 59 gives us 59 + 0.941 ≈ 59.941.

So the square root of 3593 is approximately 59.941.

• Square root: A square root is the inverse of squaring a number. Example: 4² = 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.
   
• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
   
• Decimal: If a number has a whole number and a fraction together, it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.
   
• Long division method: A technique used to find the square root of non-perfect square numbers by performing division in a systematic manner.