Square Root of 7380

The square root is the inverse of the square of the number. 7380 is not a perfect square. The square root of 7380 is expressed in both radical and exponential form. In the radical form, it is expressed as √7380, whereas (7380)^(1/2) in the exponential form. √7380 ≈ 85.909, 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 and approximation methods 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 7380 is broken down into its prime factors:

Step 1: Finding the prime factors of 7380 Breaking it down, we get 2 x 2 x 3 x 5 x 5 x 41: (2^2) x 3^1 x 5^2 x 41^1

Step 2: Now we found out the prime factors of 7380. The second step is to make pairs of those prime factors. Since 7380 is not a perfect square, the digits of the number can’t be grouped entirely in pairs. Therefore, calculating √7380 using prime factorization directly 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 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 7380, we need to group it as 80 and 73.

Step 2: Now we need to find n whose square is less than or equal to 73. We can say n is ‘8’ because 8^2 = 64 is less than 73. Now the quotient is 8, and after subtracting 64 from 73, the remainder is 9.

Step 3: Now let us bring down 80, which is the new dividend. Add the old divisor with the same number 8 + 8 = 16, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 16n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 16n × n ≤ 980. Let us consider n as 5, now 16 x 5 x 5 = 825

Step 6: Subtract 980 from 825; the difference is 155, and the quotient is 85.

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 15500.

Step 8: Now we need to find the new divisor that is 859 because 859 x 9 = 7731

Step 9: Subtracting 7731 from 15500, we get the result 7779.

Step 10: Now the quotient is 85.9

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal values, continue till the remainder is zero. So the square root of √7380 is approximately 85.91.

The approximation method is another method for finding 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 7380 using the approximation method

Step 1: Now we have to find the closest perfect square of √7380. The smallest perfect square less than 7380 is 7225, and the largest perfect square greater than 7380 is 7569. √7380 falls somewhere between 85 and 87.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (7380 - 7225) ÷ (7569 - 7225) = 155 ÷ 344 = 0.4506 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 85 + 0.45 = 85.45, so the square root of 7380 is approximately 85.45.

• Square root: A square root is the inverse of a square. For example, 9² = 81, and the inverse of the square is the square root, which is √81 = 9.
   
• 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 a principal square root.
   
• Prime factorization: Prime factorization is the expression of a number as a product of its prime factors.
   
• Approximation: Approximation is the process of finding a value that is close enough to the correct answer, usually within a tolerable error range.