Square Root of 4680

The square root is the inverse of the square of a number. 4680 is not a perfect square. The square root of 4680 is expressed in both radical and exponential form. In the radical form, it is expressed as √4680, whereas in the exponential form it is expressed as (4680)^(1/2). √4680 ≈ 68.426, 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 often 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's look at how 4680 is broken down into its prime factors.

Step 1: Finding the prime factors of 4680

Breaking it down, we get 2 x 2 x 2 x 3 x 3 x 5 x 13: 2^3 x 3^2 x 5 x 13

Step 2: We found the prime factors of 4680. The second step is to make pairs of those prime factors. Since 4680 is not a perfect square, the digits of the number can’t be grouped into pairs completely. Therefore, calculating 4680 using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. In this method, we find 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: We need to group the numbers from right to left. In the case of 4680, we group it as 80 and 46.

Step 2: Now we need to find n whose square is less than or equal to 46. The closest perfect square is 6 x 6 = 36. So the quotient is 6, after subtracting 36 from 46 the remainder is 10.

Step 3: Bring down 80, making the new dividend 1080. Add the old divisor with the same number 6 + 6 = 12, which will be our new divisor.

Step 4: We need to find the largest digit n such that 12n x n ≤ 1080. Trying n = 8, we get 128 x 8 = 1024.

Step 5: Subtract 1024 from 1080, the remainder is 56. The quotient is 68.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeroes, making it 5600.

Step 7: Now find the new divisor which is 136 (68 x 2) and find n such that 136n x n ≤ 5600. Trying n = 4, we get 1364 x 4 = 5456.

Step 8: Subtract 5456 from 5600, getting a remainder of 144. Step 9: The quotient is now 68.4 Step 10: Continue these steps until you get the desired precision.

So the square root of √4680 is approximately 68.426.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Let us learn how to find the square root of 4680 using the approximation method.

Step 1: We need to find two perfect squares between which √4680 falls. The squares of 68 (4624) and 69 (4761) are the closest perfect squares.

Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Applying the formula: (4680 - 4624) ÷ (4761 - 4624) = 56 ÷ 137 ≈ 0.409 Adding this value to the smaller integer value: 68 + 0.409 ≈ 68.409 Thus, √4680 ≈ 68.409.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: √16 = 4, because 4 x 4 = 16.

• Irrational number: An irrational number cannot be written as a simple fraction; it has a non-repeating, non-terminating decimal expansion. Example: √4680 is irrational.

• Approximation: Estimating a number close to its actual value, often used for irrational numbers.

• Perfect square: A number that is the square of an integer. Example: 16 is a perfect square because it is 4 squared.

• Long division method: A technique used to find the square root of a number by dividing it into pairs and finding the closest perfect squares.