Square Root of 4264

The square root is the inverse of the square of the number. 4264 is not a perfect square. The square root of 4264 is expressed in both radical and exponential form. In the radical form, it is expressed as √4264, whereas (4264)^(1/2) is the exponential form. √4264 ≈ 65.287, 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 4264 is broken down into its prime factors.

Step 1: Finding the prime factors of 4264 Breaking it down, we get 2 x 2 x 2 x 2 x 7 x 19: 2^4 x 7 x 19

Step 2: Now we found the prime factors of 4264. The second step is to make pairs of those prime factors. Since 4264 is not a perfect square, the digits of the number can’t be grouped in pairs evenly.

Therefore, calculating √4264 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 4264, we need to group it as 42 and 64.

Step 2: Now we need to find n whose square is closest to 42. We can say n is '6' because 6 x 6 = 36, which is less than or equal to 42. Now the quotient is 6 after subtracting 36 from 42, the remainder is 6.

Step 3: Now let us bring down 64, making the new dividend 664. Add the old divisor with the same number: 6 + 6 = 12, which will be our new divisor.

Step 4: We need to find a number n such that 12n x n ≤ 664. Let us consider n as 5, now 125 x 5 = 625.

Step 5: Subtracting 625 from 664, the difference is 39, and the quotient is 65.

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 3900.

Step 7: Find the new divisor. It will be 130.5 because 1305 x 3 = 3915, which is slightly more than 3900.

Step 8: Subtracting 3915 from 3900 gives a negative value, so adjust n to 2, making 1302 x 2 = 2604.

Step 9: Subtracting 2604 from 3900, we get a remainder of 1296.

Step 10: Continue this process until you get a satisfactory level of precision.

The square root of √4264 is approximately 65.287.

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 4264 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √4264. The smallest perfect square less than 4264 is 4225, and the largest perfect square greater than 4264 is 4356. √4264 falls somewhere between 65 and 66.

Step 2: Now apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (4264 - 4225) / (4356 - 4225) = 39 / 131 ≈ 0.298. 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 65 + 0.298 = 65.298, so the square root of 4264 is approximately 65.298.

• 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, the positive square root is more prominent due to its uses in the real world. Thus, it is known as the principal square root.
   
• Prime factorization: The process of breaking down a number into its basic prime number components.
   
• Long division method: A method used to find the square root of a number that involves dividing the number in a stepwise manner to approximate the root.