Square Root of 565

The square root is the inverse of the square of the number. 565 is not a perfect square. The square root of 565 is expressed in both radical and exponential form. In the radical form, it is expressed as √565, whereas (565)^(1/2) in the exponential form. √565 ≈ 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, for non-perfect square numbers, the long division method and approximation method are often 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 565 is broken down into its prime factors.

Step 1: Finding the prime factors of 565 Breaking it down, we get 5 × 113: 5¹ × 113¹

Step 2: We found out the prime factors of 565. Since 565 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 565 using prime factorization alone is impractical.

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 565, we need to consider it as 65 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 × 2 = 4, which is less than 5. Now the quotient is 2, and after subtracting 5 - 4, the remainder is 1.

Step 3: Bring down 65 to make it 165. Add the old divisor with the same number 2 + 2 to 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 × n ≤ 165. Let us consider n as 3, now 4 × 3 × 3 = 144.

Step 6: Subtract 165 from 144, the difference is 21, 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 2100.

Step 8: Now we need to find the new divisor that is 474 because 474 × 4 = 1896.

Step 9: Subtracting 1896 from 2100, we get the result 204.

Step 10: Now the quotient is 23.7.

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

So the square root of √565 is approximately 23.74.

The approximation method is another method for finding square roots, and 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 565 using the approximation method.

Step 1: We have to find the closest perfect squares of √565. The smallest perfect square less than 565 is 529, and the largest perfect square greater than 565 is 576. √565 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: (565 - 529) / (576 - 529) = 36 / 47 ≈ 0.766. Adding this to the smaller root gives us 23 + 0.766 = 23.766, so the square root of 565 is approximately 23.77.

• Square Root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, which 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, the positive square root is usually the one used in real-world applications, known as the principal square root.

• Prime Numbers: Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves, such as 2, 3, 5, etc.

• Decimal: A decimal is a number that consists of a whole number and a fractional part separated by a decimal point, such as 7.86 or 23.75.