Square Root of 591

The square root is the inverse of the square of a number. 591 is not a perfect square. The square root of 591 is expressed in both radical and exponential form. In the radical form, it is expressed as √591, whereas in the exponential form it is (591)^(1/2). √591 ≈ 24.3105, 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 591 is broken down into its prime factors.

Step 1: Finding the prime factors of 591 Breaking it down, we get 3 × 197: 3^1 × 197^1

Step 2: Now that we found out the prime factors of 591, the next step is to make pairs of those prime factors. Since 591 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating √591 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 numbers around 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, group the numbers from right to left. In the case of 591, we group it as 91 and 5.

Step 2: Find 'n' such that n^2 is less than or equal to 5. Here, n is 2 because 2^2 = 4. Now the quotient is 2, and after subtracting 4 from 5, the remainder is 1.

Step 3: Bring down 91, making it the new dividend. Add the old divisor (2) with itself to get 4 as the new divisor.

Step 4: Find 'n' such that 4n × n is less than or equal to 191. Let n be 4, then 44 × 4 = 176.

Step 5: Subtract 176 from 191 to get 15, and the quotient becomes 24.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeros to make it 1500.

Step 7: Find the new divisor, which is 481, as 481 × 3 = 1443.

Step 8: Subtract 1443 from 1500 to get 57.

Step 9: Continue this process until the desired level of precision is reached. The square root of √591 is approximately 24.31.

The approximation method is another way to find square roots, and it is an easy method for estimating the square root of a given number. Now let us learn how to find the square root of 591 using the approximation method.

Step 1: Find the closest perfect squares around 591. The closest perfect square below 591 is 576 (24^2), and the closest perfect square above is 625 (25^2). √591 falls between 24 and 25.

Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) Using the formula: (591 - 576) / (625 - 576) = 15 / 49 ≈ 0.306 Adding this to the smaller square root gives us 24 + 0.306 ≈ 24.31, so the square root of 591 is approximately 24.31.

• Square root: A square root is the inverse operation of squaring a number. For example, 5^2 = 25, and the inverse is √25 = 5.

• Irrational number: An irrational number cannot be written as a simple fraction or in the form p/q, where p and q are integers and q ≠ 0.

• Principal square root: The principal square root is the non-negative square root of a number. For example, the principal square root of 25 is 5.

• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.

• Long division method: A step-by-step approach to finding the square root of a number, particularly useful for non-perfect squares.