Square Root of 7936

The square root is the inverse of the square of the number. 7936 is not a perfect square. The square root of 7936 is expressed in both radical and exponential form. In the radical form, it is expressed as √7936, whereas (7936)^(1/2) in the exponential form. √7936 ≈ 89.056, 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 7936 is broken down into its prime factors.

Step 1: Finding the prime factors of 7936 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 13 x 13: 2^6 x 13^2

Step 2: Now we found out the prime factors of 7936. The second step is to make pairs of those prime factors. Since 7936 is not a perfect square, the digits of the number can’t be paired completely. Therefore, calculating 7936 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 7936, we need to group it as 36 and 79.

Step 2: Now we need to find n whose square is close to 79. We can say n as ‘8’ because 8 x 8 = 64, which is less than 79. Now the quotient is 8, and after subtracting 64 from 79, the remainder is 15.

Step 3: Bring down 36, making the new dividend 1536. Add the old divisor with the same number 8 + 8 = 16, which will be our new divisor.

Step 4: The next step is finding 16n × n ≤ 1536. Let us consider n as 9, now 16 x 9 = 1449.

Step 5: Subtracting 1449 from 1536 gives a remainder of 87, and the quotient is 89.

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, making it 8700.

Step 7: We continue with the long division process until we get a sufficiently accurate decimal approximation. The square root of √7936 ≈ 89.056.

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 7936 using the approximation method.

Step 1: We have to find the closest perfect square of √7936. The smallest perfect square close to 7936 is 7921 (89^2) and the largest is 8100 (90^2). √7936 falls between 89 and 90.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (7936 - 7921) / (8100 - 7921) = 15 / 179 ≈ 0.084 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 89 + 0.084 = 89.084, so the square root of 7936 is approximately 89.084.

• Square root: A square root is the inverse of a square. For example, 4^2 = 16, and the square root is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written as a simple fraction; it has a non-repeating, non-terminating decimal.
   
• Principal square root: The principal square root is the positive square root of a number, commonly used due to its practical applications.
   
• Decimal: A decimal is a number that contains a whole number and a fractional part separated by a decimal point, such as 7.86, 8.65, and 9.42.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 x 3 x 3.