Square Root of 782

The square root is the inverse of the square of the number. 782 is not a perfect square. The square root of 782 is expressed in both radical and exponential form. In the radical form, it is expressed as √782, whereas (782)^(1/2) in the exponential form. √782 = 27.96426, 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 782 is broken down into its prime factors.

Step 1: Finding the prime factors of 782 Breaking it down, we get 2 x 17 x 23.

Step 2: Now we found out the prime factors of 782. The second step is to make pairs of those prime factors. Since 782 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating the square root of 782 using prime factorization 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 782, we need to group it as 82 and 7.

Step 2: Now we need to find n whose square is less than or equal to 7. We can say n as ‘2’ because 2 x 2 = 4 is less than 7. Now the quotient is 2, and after subtracting 4 from 7, the remainder is 3.

Step 3: Now let us bring down 82, which is the new dividend. Add the old divisor with the same number 2 + 2 = 4, which will be our new divisor.

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

Step 5: The next step is finding 4n × n ≤ 382. Let us consider n as 9, now 49 x 9 = 441.

Step 6: Subtracting gives us a negative remainder, so try n as 8, 48 x 8 = 384.

Step 7: Subtract 382 from 384, and the difference is negative, indicating 8 is still too high, so try 47 x 8 = 376, giving a remainder of 6.

Step 8: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 600.

Step 9: Find the new divisor. Since 479 x 9 = 4311 is too high, try 478 x 8 = 3824.

Step 10: Subtracting gives a remainder of 176. Continue with these steps.

So the square root of √782 is approximately 27.96.

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 782 using the approximation method.

Step 1: Now we have to find the closest perfect square of √782.

The smallest perfect square less than 782 is 729, and the largest perfect square greater than 782 is 841. √782 falls somewhere between 27 and 29.

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 (782 - 729) / (841 - 729) = 53 / 112 ≈ 0.4732.

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 27 + 0.4732 ≈ 27.97.

So, the square root of 782 is approximately 27.97.

• 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, which 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: The process of breaking down a number into its smallest prime factors. For example, the prime factorization of 782 is 2 x 17 x 23.

• Long division method: A method used for finding the square root of non-perfect squares by grouping numbers and using division to approximate the square root.