Square Root of 694

The square root is the inverse of the square of the number. 694 is not a perfect square. The square root of 694 is expressed in both radical and exponential form. In the radical form, it is expressed as √694, whereas 694^(1/2) in the exponential form. √694 ≈ 26.342, 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 694 is broken down into its prime factors.

Step 1: Finding the prime factors of 694

Breaking it down, we get 2 x 347. 694 = 2^1 x 347^1

Step 2: Now we found out the prime factors of 694. The second step is to make pairs of those prime factors. Since 694 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 694 using prime factorization is not feasible 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, we need to group the numbers from right to left. In the case of 694, we write it as 6 and 94.

Step 2: Now, find n whose square is less than or equal to 6. We take n as ‘2’ because 2 x 2 = 4, which is less than 6. The quotient is 2, and after subtracting 4 from 6, the remainder is 2.

Step 3: Bring down 94, making the new dividend 294. 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 the sum of the previous quotient and double it. We have 4n as the new divisor; we need to find n such that 4n x n ≤ 294.

Step 5: Let n be 6, so 46 x 6 = 276, which is less than 294.

Step 6: Subtract 276 from 294; the difference is 18, and the quotient becomes 26.

Step 7: 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 1800.

Step 8: Now find the new divisor, which is 526. When 526 is multiplied by 3, we get 1578.

Step 9: Subtracting 1578 from 1800 gives us 222.

Step 10: The quotient so far is 26.3.

Step 11: Continue these steps until we get two decimal places or until the remainder is zero.

So the square root of √694 is approximately 26.34.

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 694 using the approximation method.

Step 1: Find the closest perfect squares around √694. The smallest perfect square less than 694 is 676, and the largest perfect square greater than 694 is 729. √694 falls somewhere between 26 and 27.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (694 - 676) / (729 - 676) = 18 / 53 ≈ 0.34 Using the formula, we identified the decimal point of our square root. Adding this to the whole number gives us 26 + 0.34 = 26.34, so the square root of 694 is approximately 26.34.

• 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 more commonly used in practical applications. This is known as the principal square root.

• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 12 is 2^2 x 3.

• Long division method: A technique used to divide numbers to find the square root of non-perfect squares involving repeated division steps.