Square Root of 562

The square root is the inverse of the square of a number. 562 is not a perfect square. The square root of 562 is expressed in both radical and exponential form. In the radical form, it is expressed as √562, whereas (562)^(1/2) in the exponential form. √562 ≈ 23.70654, 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 the 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 562 is broken down into its prime factors.

Step 1: Finding the prime factors of 562 Breaking it down, we get 2 x 281: 2¹ x 281¹

Step 2: Now we found out the prime factors of 562. The second step is to make pairs of those prime factors. Since 562 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 562 using prime factorization to find its square root 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, we need to group the numbers from right to left. In the case of 562, we need to group it as 62 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, which is less than 5. Now the quotient is 2, and after subtracting 4 from 5, the remainder is 1.

Step 3: Now let us bring down 62, which is the new dividend. Add the old divisor (2) with itself to get 4, which will serve as the first part of our new divisor.

Step 4: The new divisor will be 4n. We need to find the value of n such that 4n x n is less than or equal to 162.

Step 5: Let n be 3. Then 43 x 3 = 129.

Step 6: Subtract 129 from 162; the difference is 33. The quotient is now 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 3300.

Step 8: We find that 474 x 7 = 3318.

Step 9: Subtracting 3318 from 3300 gives us a negative number, so we try 473 x 6 = 2838.

Step 10: Subtract 2838 from 3300, resulting in 462.

Step 11: Since we desire precision, continue the steps until achieving the desired decimal places.

Thus, the square root of √562 is approximately 23.706.

The approximation method is another method for finding square roots. It is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 562 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √562. The smallest perfect square less than 562 is 529, and the largest perfect square more than 562 is 576. √562 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).

Using the formula (562 - 529) / (576 - 529) = 33 / 47 ≈ 0.702 Using this approximation, we identified the decimal portion of our square root. The next step is adding the integer part: 23 + 0.702 = 23.702.

Therefore, the approximate square root of 562 is 23.702.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: 4² = 16, and the inverse operation is the square root, √16 = 4.

• Irrational number: An irrational number cannot be written as a simple fraction or ratio of integers. Its decimal form is non-terminating and non-repeating.

• Principal square root: The principal square root of a number is its non-negative square root. For example, the principal square root of 9 is 3.

• Long division method: A technique used to find the square root of a non-perfect square by dividing and averaging.

• Approximation method: A method used to estimate the square root of a non-perfect square by comparing it to nearby perfect squares.