Square Root of 568

The square root is the inverse of squaring a number. 568 is not a perfect square. The square root of 568 is expressed in both radical and exponential form. In radical form, it is expressed as √568, whereas in exponential form, it is expressed as (568)^(1/2). √568 ≈ 23.83275, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

The prime factorization method is employed for perfect square numbers. However, for non-perfect square numbers, methods such as the long-division method and the approximation method are used. Let us now learn about these 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 568 is broken down into its prime factors:

Step 1: Finding the prime factors of 568 Breaking it down, we get 2 x 2 x 2 x 71: 2^3 x 71

Step 2: Now we have found the prime factors of 568. Since 568 is not a perfect square, its digits cannot be grouped into pairs. Therefore, calculating the square root of 568 using prime factorization involves approximating the pairs, which is not straightforward.

The long division method is particularly useful for non-perfect square numbers. In this method, we should find the closest perfect square number to 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 568, we need to group it as 68 and 5.

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

Step 3: Now bring down 68, making the new dividend 168. Add the old divisor with the same number: 2 + 2 = 4, making 4 the new divisor.

Step 4: The new divisor will be 4n. We need to find n such that 4n x n ≤ 168. Let us consider n as 3, now 43 x 3 = 129.

Step 5: Subtract 129 from 168, resulting in a remainder of 39, 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 3900.

Step 7: Find the new divisor, which is 476. Now 476 x 8 = 3808.

Step 8: Subtracting 3808 from 3900 gives a result of 92.

Step 9: Now the quotient is 23.8.

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

So, the square root of √568 ≈ 23.83.

The approximation method is an easy way to find the square roots of a given number. Now let us learn how to find the square root of 568 using the approximation method.

Step 1: We need to find the closest perfect squares to √568. The smallest perfect square less than 568 is 529 (√529 = 23), and the largest perfect square greater than 568 is 576 (√576 = 24). √568 is between 23 and 24.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (568 - 529) / (576 - 529) = 39 / 47 ≈ 0.83. Adding the integer part to the decimal: 23 + 0.83 = 23.83, so the square root of 568 is approximately 23.83.

• Square root: A square root is the inverse operation of squaring a number. For example, 4² = 16, and √16 = 4.

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

• Principal square root: A number has both positive and negative square roots, but the principal square root is the non-negative one, which is commonly used in practical applications.

• Prime factorization: Breaking down a number into its basic prime factors. It is useful in various mathematical calculations.

• Decimal number: A number that has a whole number and a fractional part, separated by a decimal point. For example: 7.86, 8.65, and 9.42 are decimals.