Square Root of 4633

The square root is the inverse of the square of a number. 4633 is not a perfect square. The square root of 4633 is expressed in both radical and exponential form. In the radical form, it is expressed as √4633, whereas (4633)^(1/2) in the exponential form. √4633 ≈ 68.072, 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 4633 is broken down into its prime factors.

Step 1: Finding the prime factors of 4633 Breaking it down, we find that 4633 = 13 x 357. Since 357 is also not a prime number, we break it down further into 3 x 119, and 119 into 7 x 17. Hence, the prime factorization of 4633 is 13 x 3 x 7 x 17.

Step 2: Since 4633 is not a perfect square, calculating its square root using prime factorization is not straightforward. We proceed with other methods.

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 4633, we need to group it as 46 and 33.

Step 2: Now we need to find n whose square is less than or equal to 46. We choose n as ‘6’ because 6 x 6 = 36, which is less than 46. Now the quotient is 6, and after subtracting 36 from 46, the remainder is 10.

Step 3: Bring down 33, making the new dividend 1033. Double the quotient to get the new divisor, which is 12.

Step 4: Find a digit n such that 12n x n is less than or equal to 1033. We choose n as 8, because 128 x 8 = 1024, which is less than 1033.

Step 5: Subtract 1024 from 1033 to get a remainder of 9.

Step 6: 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 900.

Step 7: Continue the process to find additional digits after the decimal point.

So the square root of √4633 ≈ 68.072

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 4633 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √4633. The closest perfect squares to 4633 are 4624 (68^2) and 4761 (69^2). √4633 falls somewhere between 68 and 69.

Step 2: Now we need to apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). Using the formula (4633 - 4624) / (4761 - 4624) = 9 / 137 ≈ 0.066. Adding this value to the smaller perfect square's root: 68 + 0.066 ≈ 68.066

So the square root of 4633 is approximately 68.072.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, as 4 x 4 = 16.

• Irrational number: An irrational number cannot be written as a simple fraction, meaning its decimal form is non-repeating and non-terminating.

• Factorization: The process of breaking down a number into its constituent prime factors, such as breaking down 4633 into 13, 3, 7, and 17.

• Approximation: A numerical value close to the actual value, often used when the exact value cannot be easily calculated.

• Long division: A method used to divide larger numbers and to find square roots of non-perfect squares in a step-by-step manner.