Square Root of 684

The square root is the inverse of the square of the number. 684 is not a perfect square. The square root of 684 is expressed in both radical and exponential form. In radical form, it is expressed as √684, whereas \(684^{1/2}\) is the exponential form. √684 ≈ 26.15339, 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 684 is broken down into its prime factors.

Step 1: Finding the prime factors of 684 Breaking it down, we get \(2 \times 2 \times 3 \times 3 \times 19\).

Step 2: Now we found out the prime factors of 684. The second step is to make pairs of those prime factors. Since 684 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √684 using prime factorization directly 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 684, we need to group it as 84 and 6.

Step 2: Now we need to find n whose square is 6. We can say n as '2' because \(2^2\) is lesser than or equal to 6. Now the quotient is 2. Subtracting 4 from 6 gives a remainder of 2.

Step 3: Now let us bring down 84, which is the new dividend. 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 dividend and quotient. Now we get 4n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding \(4n \times n ≤ 284\). Let's consider n as 6, now \(46 \times 6 = 276\).

Step 6: Subtract 276 from 284, the difference is 8, and the quotient is 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 800.

Step 8: Now we need to find the new divisor. The new divisor is 523 because \(523 \times 1 = 523\).

Step 9: Subtracting 523 from 800 gives us the result 277.

Step 10: Now the quotient is 26.1.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √684 ≈ 26.15.

The approximation method is another method for finding the 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 684 using the approximation method.

Step 1: Now we have to find the closest perfect square of √684. The smallest perfect square less than 684 is 676 (which is \(26^2\)) and the largest perfect square greater than 684 is 729 (which is \(27^2\)). Therefore, √684 falls somewhere between 26 and 27.

Step 2: Now we need to apply the formula: \((\text{Given number - smallest perfect square}) / (\text{Greater perfect square - smallest perfect square})\). Using the formula \((684 - 676) / (729 - 676) \approx 0.15339\). 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 \(26 + 0.15339 \approx 26.15339\), so the square root of 684 is approximately 26.15339.

• 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.

• Radical: A radical is a symbol (√) used to indicate the square root or nth roots of a number.

• Exponential Form: Exponential form is a way to express a number using a base and an exponent. For example, \(a^b\) is an exponential form where 'a' is the base and 'b' is the exponent.

• Perfect Square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because it is \(4^2\).