Square Root of 564

The square root is the inverse of the square of a number. 564 is not a perfect square. The square root of 564 is expressed in both radical and exponential form. In radical form, it is expressed as √564, whereas (564^)(1/2) in exponential form. √564 ≈ 23.7487, 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 564 is broken down into its prime factors.

Step 1: Finding the prime factors of 564. Breaking it down, we get 2 x 2 x 3 x 47: 2² x 3 x 47.

Step 2: Now we found out the prime factors of 564. The second step is to make pairs of those prime factors. Since 564 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 564 using prime factorization is not straightforward.

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 564, we need to group it as 64 and 5.

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

Step 3: Now let us bring down 64, making the new dividend 164. 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 4n. We need to find n such that 4n x n ≤ 164. Let us consider n as 3, now 43 x 3 = 129.

Step 5: Subtract 129 from 164, the difference is 35, and the quotient is 23.

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 3500.

Step 7: Now we need to find the new divisor, which is 474 because 474 x 7 = 3318.

Step 8: Subtracting 3318 from 3500, we get the result 182.

Step 9: Now the quotient is 23.7 Step 10: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.

So the square root of √564 is approximately 23.75.

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 564 using the approximation method

.Step 1: Now we have to find the closest perfect squares of √564. The closest perfect square less than 564 is 529 (23²) and more than 564 is 576 (24²). √564 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 (564 - 529) / (576 - 529) = 35/47 ≈ 0.7447

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 + 0.7447 ≈ 23.7447, so the square root of 564 is approximately 23.7447.

• Square root: A square root is the inverse of a square. For 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 expressed 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 is more prominent due to its uses in the real world. That is the reason it is also known as the principal square root.

• Approximation: A method of estimating the value of a mathematical expression when an exact answer is not needed or is difficult to obtain.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.