Square Root of 4689

The square root is the inverse of the square of the number. 4689 is not a perfect square. The square root of 4689 can be expressed in both radical and exponential forms. In the radical form, it is expressed as √4689, whereas (4689)^(1/2) in the exponential form. √4689 ≈ 68.4653, 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 long division and approximation methods 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 4689 is broken down into its prime factors:

Step 1: Finding the prime factors of 4689

Breaking it down, we get 3 x 3 x 521: 3^2 x 521

Step 2: Now we found the prime factors of 4689. Since 4689 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √4689 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 4689, we need to group it as 89 and 46.

Step 2: Now we need to find n whose square is ≤ 46. We can say n as ‘6’ because 6 x 6 = 36, which is less than 46. Subtract 36 from 46, and the remainder is 10.

Step 3: Bring down 89, which is the new dividend. Add the old divisor with the same number 6 + 6, we get 12, which will be our new divisor.

Step 4: The new divisor will have a placeholder for the next digit. Now we get 12_ as the new divisor, and we need to find the value of n.

Step 5: Find a digit for n where 12n x n ≤ 1089. Let us consider n as 8, now 128 x 8 = 1024.

Step 6: Subtract 1024 from 1089, the difference is 65, and the quotient is 68.

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 6500.

Step 8: Find the new divisor, which becomes 136 because 136 x 5 = 6800 is the closest without exceeding 6500.

Step 9: Subtracting 6800 from 6500 gives the remainder.

Step 10: The quotient becomes 68.46.

Step 11: Continue these steps until we get two numbers after the decimal point or when repeating steps yields a repeating pattern.

So, the square root of √4689 is approximately 68.4653.

The approximation method is another method for finding square roots. It is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 4689 using the approximation method:

Step 1: Identify the closest perfect squares to √4689. The smallest perfect square less than 4689 is 4624, and the largest perfect square more than 4689 is 4761. √4689 falls somewhere between 68 and 69.

Step 2: Apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (4689 - 4624) / (4761 - 4624) = 65 / 137 ≈ 0.4745. Adding this decimal to the lower guess, we get 68 + 0.4745 = 68.4745. Therefore, √4689 ≈ 68.4745.

• 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 used in practical applications, known as the principal square root.

• Prime factorization: The expression of a number as a product of its prime factors, such as 4689 = 3 x 3 x 521.

• Decimal approximation: The process of estimating an irrational number using decimal points for practical usage, such as √4689 ≈ 68.4653.