Square Root of 788

The square root is the inverse of the square of the number. 788 is not a perfect square. The square root of 788 is expressed in both radical and exponential form. In the radical form, it is expressed as √788, whereas (788)^(1/2) in the exponential form. √788 ≈ 28.072, 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, for non-perfect square numbers, the long division method and approximation method are commonly 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 788 is broken down into its prime factors.

Step 1: Finding the prime factors of 788 Breaking it down, we get 2 x 2 x 197.

Step 2: Now we found out the prime factors of 788. Since 788 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating √788 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 788, we need to group it as 88 and 7.

Step 2: Now we need to find n whose square is closest to or less than 7. We can choose n as '2' because 2 x 2 = 4 is less than 7. Now the quotient is 2 and the remainder is 3.

Step 3: Bring down 88, making the new dividend 388. Add the old divisor with the same number, 2 + 2, to get 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find the value of n such that 4n x n is less than or equal to 388.

Step 5: Let n be 7, then 47 x 7 = 329.

Step 6: Subtract 329 from 388 to get the remainder 59, and the quotient becomes 27.

Step 7: Since the dividend is less than the divisor now, we add a decimal point and two zeroes to the dividend, making it 5900.

Step 8: Find the new divisor: 274, because 274 x 10 = 2740, which fits into 5900.

Step 9: Subtract 2740 from 5900 to get 3160.

Step 10: The quotient now is 28.0. Continue with these steps until you achieve the desired decimal precision.

So the square root of √788 is approximately 28.072.

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 788 using the approximation method.

Step 1: Find the closest perfect squares around 788.

The smallest perfect square is 784, and the largest perfect square is 841. √788 falls somewhere between 28 and 29.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (788 - 784) / (841 - 784) = 4 / 57 ≈ 0.070. Add this to the lower estimate: 28 + 0.070 = 28.070, so the square root of 788 is approximately 28.070.

• 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 is often used in practical applications.

• Prime factorization: Breaking down a number into its basic prime number factors. For example, 788 = 2 x 2 x 197.

• Decimal: A number that contains a whole number and a fraction in a single number, for example: 28.072, 3.14, and 6.283 are decimals.