Square Root of 7361

The square root is the inverse of the square of a number. 7361 is not a perfect square. The square root of 7361 is expressed in both radical and exponential form. In the radical form, it is expressed as √7361, whereas 7361^(1/2) in the exponential form. √7361 ≈ 85.792, 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 7361 is broken down into its prime factors.

Step 1: Finding the prime factors of 7361 Breaking it down, we get 13 x 13 x 43: 13^2 x 43

Step 2: Now that we found the prime factors of 7361, the second step is to make pairs of those prime factors. Since 7361 is not a perfect square, the digits of the number can’t be grouped into pairs. Therefore, calculating 7361 using prime factorization is impossible.

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 7361, we need to group it as 73 and 61.

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

Step 3: Now let us bring down 61, which is the new dividend. Add the old divisor with the same number 8 + 8, we get 16, which will be our new divisor.

Step 4: The new divisor will be 160. Now we need to find a digit n such that 160n x n ≤ 961. Let us take n = 5, then 1605 x 5 = 8025.

Step 5: Subtract 961 - 8025, the difference is 1596.

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 159600.

Step 7: We repeat the process to calculate the next digits after the decimal. So the square root of √7361 ≈ 85.792

The approximation method is another way to find 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 7361 using the approximation method.

Step 1: Now we have to find the closest perfect square to √7361. The smallest perfect square greater than 7361 is 7396, and the largest perfect square less than 7361 is 7225. √7361 falls somewhere between 85 and 86.

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 (7361 - 7225) ÷ (7396 - 7225) ≈ 0.792 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 + 0.792 ≈ 85.792, so the square root of 7361 is approximately 85.792.

• 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 usually the positive square root that is of more prominence due to its uses in the real world. That is why it is also known as the principal square root.
   
• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors.
   
• Long division method: The long division method is a systematic way to find the square root of a number by dividing and averaging to get closer to the actual root.