Square Root of 572

The square root is the inverse of the square of the number. 572 is not a perfect square. The square root of 572 is expressed in both radical and exponential form. In the radical form, it is expressed as √572, whereas in the exponential form it is expressed as (572)^(1/2). √572 = 23.91652, 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:

• 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 572 is broken down into its prime factors.

Step 1: Finding the prime factors of 572 Breaking it down, we get 2 × 2 × 11 × 13: 2^2 × 11 × 13

Step 2: Now that we have found the prime factors of 572, the second step is to make pairs of those prime factors. Since 572 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √572 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 572, we need to group it as 72 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 is 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 72, which is the new dividend. Add the old divisor with itself, 2 + 2, to get 4, which will be our new divisor.

Step 4: We have 4n as the new divisor, and we need to find the value of n such that 4n × n is less than or equal to 172. Let's consider n as 4. Now 44 × 4 = 176, which is greater than 172, so we choose n as 3.

Step 5: Subtract 144 (43 × 3) from 172, and the difference is 28. 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 2800.

Step 7: Now we need to find the new divisor, which is 46 because 463 × 3 = 1389.

Step 8: Subtract 1389 from 2800 to get the result of 1411.

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

So the square root of √572 is approximately 23.92.

The 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 572 using the approximation method.

Step 1: Now we have to find the closest perfect square to √572. The smallest perfect square less than 572 is 529, and the largest perfect square greater than 572 is 576. √572 falls somewhere between 23 and 24.

Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula, (572 - 529) ÷ (576 - 529) = 43 ÷ 47 ≈ 0.9149 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 + 0.9149 = 23.9149.

So the square root of 572 is approximately 23.9149.

• 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 the principal square root.

• Prime factorization: Writing a number as a product of its prime factors. Example: The prime factorization of 28 is 2 × 2 × 7.

• 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.