Square Root of 561

The square root is the inverse of the square of the number. 561 is not a perfect square. The square root of 561 is expressed in both radical and exponential form. In the radical form, it is expressed as √561, whereas (561)^(1/2) in the exponential form. √561 ≈ 23.685, 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:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 561 is broken down into its prime factors.

Step 1: Finding the prime factors of 561 Breaking it down, we get 3 x 11 x 17: 3¹ x 11¹ x 17¹

Step 2: Now we found out the prime factors of 561. The second step is to make pairs of those prime factors. Since 561 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 561 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 561, we need to group it as 61 and 5.

Step 2: Now we need to find n whose square is 5. We can say n as ‘2’ because 2 x 2 is lesser than or equal to 5. Now the quotient is 2. After subtracting 4 from 5, the remainder is 1.

Step 3: Now let us bring down 61, which is the new dividend. Add the old divisor with the same number 2 + 2; we get 4, which will be our new divisor.

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

Step 5: The next step is finding 4n x n ≤ 161. Let us consider n as 3; now 4 x 3 x 3 = 129.

Step 6: Subtract 129 from 161; the difference is 32, and the quotient is 23.

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 3200.

Step 8: Now we need to find the new divisor that is 6 because 473 x 6 = 2838.

Step 9: Subtracting 2838 from 3200, we get the result 362.

Step 10: Now the quotient is 23.6.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √561 is approximately 23.68.

Approximation method is another method for finding the 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 561 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √561. The smallest perfect square less than 561 is 529, and the largest perfect square more than 561 is 576. √561 falls somewhere between 23 and 24.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Going by the formula (561 - 529) / (576 - 529) = 32 / 47 ≈ 0.68. 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 23 + 0.68 ≈ 23.68, so the square root of 561 is approximately 23.68.

• Square root: A square root is the inverse of a square. 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.

• Principal square root: A number has both positive and negative square roots; however, it is always the 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: Prime factorization is the process of writing a number or algebraic expression as a product of prime numbers.

• Long division method: A method used to find the square root of a number by dividing it into groups of digits and using a series of division and subtraction steps.