Square Root of 678

The square root is the inverse of the square of a number. 678 is not a perfect square. The square root of 678 is expressed in both radical and exponential form. In radical form, it is expressed as √678, whereas in exponential form, it is expressed as (678)^(1/2). √678 ≈ 26.0384, 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 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. Let us look at how 678 is broken down into its prime factors:

Step 1: Finding the prime factors of 678. Breaking it down, we get 2 x 3 x 113: 2^1 x 3^1 x 113^1.

Step 2: Now we have found the prime factors of 678. The second step is to make pairs of those prime factors. Since 678 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 678 using prime factorization is not straightforward for finding its square root.

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, group the numbers from right to left. In the case of 678, group it as 78 and 6.

Step 2: Now find n whose square is 6 or less. We can say n is '2' because 2 x 2 = 4, which is less than or equal to 6. Now the quotient is 2, and after subtracting 4 from 6, the remainder is 2.

Step 3: Bring down 78, making the new dividend 278. 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 40 (since 4 is the first part of it). We need to find n such that 40n x n ≤ 278. Let us consider n as 6, now 406 x 6 = 2436.

Step 5: Since 2436 is larger than 278, try n as 5, now, 405 x 5 = 2025, which is less than 278.

Step 6: Subtract 2025 from 278, the difference is 253, and the quotient is 25.

Step 7: Since the dividend is less than the divisor, add a decimal point and two zeros to the dividend. The new dividend is 25300.

Step 8: Find the new divisor, which is 520. Find n such that 520n x n is less than or equal to 25300.

Step 9: Continue this process until you get the required decimal places.

So the square root of √678 is approximately 26.038.

The approximation method is another method for finding square roots. It is an easy method to estimate the square root of a given number. Let us learn how to find the square root of 678 using the approximation method.

Step 1: Find the closest perfect squares around √678. The closest perfect square less than 678 is 625, and the closest perfect square greater than 678 is 729. Thus, √678 falls between 25 and 27.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (678 - 625) / (729 - 625) = 53 / 104 ≈ 0.51. Step 3: Add this decimal to the smaller integer value, 25 + 0.51 = 25.51.

This gives an approximate value of √678, but refinement processes give a closer approximation as 26.038.

• Square root: A square root is the inverse of a square. For example: 4^2 = 16, and the inverse of the square is the square root, so √16 = 4.
   
• Irrational number: An irrational number 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, typically, the positive square root is used in real-world applications. This is known as the principal square root.
   
• Prime factorization: The process of breaking down a number into its basic building blocks of prime numbers.
   
• Long division method: A step-by-step approach to finding the square root of a non-perfect square number using division and subtraction.