Square Root of 7200

The square root is the inverse of the square of the number. 7200 is not a perfect square. The square root of 7200 is expressed in both radical and exponential form. In the radical form, it is expressed as √7200, whereas (7200)^(1/2) in the exponential form. √7200 ≈ 84.8528, 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:

• 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 7200 is broken down into its prime factors.

Step 1: Finding the prime factors of 7200 Breaking it down, we get 2^4 × 3^2 × 5^2: 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5

Step 2: Now we found the prime factors of 7200. The second step is to make pairs of those prime factors. Since 7200 is not a perfect square, therefore the digits of the number can’t be grouped in complete pairs. Therefore, calculating 7200 using prime factorization results in √7200 = 2^2 × 3 × 5 × √2 = 60√2 ≈ 84.8528.

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 7200, we need to group it as 00 and 72.

Step 2: Now we need to find n whose square is closest to 72. We can say n is '8' because 8 × 8 = 64 is less than 72. Now the quotient is 8, subtracting 64 from 72 gives a remainder of 8.

Step 3: Bring down 00, making the new dividend 800.

Step 4: Double the quotient, which is 8, to get 16. This will be part of our new divisor, which will be 16n.

Step 5: Find a digit n such that 16n × n is less than or equal to 800. Let n be 4, then 164 × 4 = 656.

Step 6: Subtract 656 from 800, the remainder is 144, and add decimal point to quotient, making it 84.

Step 7: Bring down the next pair of zeros, making the new dividend 14400.

Step 8: Double the quotient (84), making it 168, and find a digit n such that 168n × n is less than or equal to 14400. Let n be 8, then 1688 × 8 = 13504.

Step 9: Subtract 13504 from 14400, the remainder is 896.

Step 10: Continue doing these steps until you get the desired number of decimal places. So, the approximate square root of √7200 is 84.8528.

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 7200 using the approximation method.

Step 1: Now we have to find the closest perfect square of √7200. The smallest perfect square less than 7200 is 7056, and the largest perfect square greater than 7200 is 7225. √7200 falls somewhere between 84 and 85.

Step 2: Apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula (7200 - 7056) ÷ (7225 - 7056) ≈ 0.8528. 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 84 + 0.8528 = 84.8528. So, the square root of 7200 is approximately 84.8528.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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: Breaking down a number into its basic building blocks, which are prime numbers. For example, the prime factorization of 7200 is 2^4 × 3^2 × 5^2.
   
• Decimal approximation: A method to estimate the value of a square root to a certain number of decimal places for practical use. For example, √7200 ≈ 84.8528.