Square Root of 8464

The square root is the inverse of the square of the number. 8464 is a perfect square. The square root of 8464 is expressed in both radical and exponential form. In the radical form, it is expressed as √8464, whereas (8464)^(1/2) in the exponential form. √8464 = 92, which is a rational number because it can 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. The long division method and approximation method can also be used to find square roots. 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 8464 is broken down into its prime factors.

Step 1: Finding the prime factors of 8464 Breaking it down, we get 2 x 2 x 2 x 2 x 13 x 13: 2^4 x 13^2

Step 2: Now we found out the prime factors of 8464. The next step is to make pairs of those prime factors. Since 8464 is a perfect square, we can group the digits into pairs. Therefore, calculating 8464 using prime factorization is possible.

The long division method is particularly useful for non-perfect square numbers, but it can also confirm results for perfect squares. 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 8464, we group it as 84 and 64.

Step 2: Find a number whose square is less than or equal to 84. This is 9 because 9 x 9 = 81.

Step 3: The quotient is 9. Subtract 81 from 84, the remainder is 3. Bring down the next pair, 64, to make the new dividend 364.

Step 4: Double the quotient and use it as part of the new divisor. So, 9 x 2 = 18.

Step 5: Find a digit n such that 18n x n is less than or equal to 364. Here, n is 2, because 182 x 2 = 364.

Step 6: Subtract 364 from 364 to get a remainder of 0. Since there's no remainder, the quotient is 92, confirming √8464 = 92.

The approximation method is an easy method for finding the square roots of non-perfect squares but can confirm perfect squares by showing proximity to calculated roots.

Step 1: Find the closest perfect square numbers around 8464. Since 8464 is a perfect square, its square root is exactly between 91 and 93.

Step 2: Verify by calculating: 91^2 = 8281 and 93^2 = 8649. Since 8464 is exactly between these, it confirms √8464 = 92.

• Square root: A square root is the inverse of a square. Example: 9^2 = 81, and the inverse of the square is the square root, that is √81 = 9.
   
• Rational number: A rational number is a number that can be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Perfect square: A number that is the square of an integer. For example, 64 is a perfect square because it is 8^2.
   
• Prime factorization: The process of expressing a number as a product of its prime factors. For example, the prime factorization of 8464 is 2^4 x 13^2.
   
• Long division method: A method used to find the square root of a number by dividing and averaging. This is particularly useful for non-perfect squares but can confirm perfect squares.