Square Root of 13456

The square root is the inverse of the square of a number. 13456 is not a perfect square. The square root of 13456 is expressed in both radical and exponential form. In radical form, it is expressed as √13456, whereas in exponential form it is expressed as (13456)^(1/2). √13456 ≈ 116.007, 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 method and approximation method are used. Let us now learn the following methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 13456 is broken down into its prime factors.

Step 1: Finding the prime factors of 13456 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 421: 2⁵ x 421

Step 2: Now we found out the prime factors of 13456. The second step is to make pairs of those prime factors. Since 13456 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 13456 using prime factorization to find its square root is not feasible.

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 13456, we need to group it as 56, 34, and 1.

Step 2: Now we need to find n whose square is less than or equal to 13. We can say n is '3' because 3 x 3 = 9, which is less than 13. Now the quotient is 3, and after subtracting 9 from 13, the remainder is 4.

Step 3: Bring down 34 to make the new dividend 434. Add the old divisor (3) to itself to get 6, which will be the start of our new divisor.

Step 4: Now we need to find n such that 6n x n ≤ 434. Let n be 7, then 67 x 7 = 469, which is greater than 434, so n should be 6.

Step 5: Subtract 396 from 434, leaving a remainder of 38. Now the quotient is 36. Step 6: Bring down 56 to make the new dividend 3856.

Step 7: The new divisor starts with 72. Repeat the process until two decimal places are achieved. The square root of 13456 is approximately 116.007.

The approximation method is another method for finding square roots. It is an easy way to find the square root of a given number. Now let us learn how to find the square root of 13456 using the approximation method.

Step 1: Now we have to find the closest perfect square numbers to √13456. The closest perfect squares are 13456 and 14400. √13456 falls between 116 and 120.

Step 2: Now we need to apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Applying the formula: (13456 - 13456) / (14400 - 13456) = 0 Using the formula, the square root approximation is 116.

• Square root: A square root is the inverse of a square. Example: 4² = 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 commonly used due to its practical applications.

• Prime factorization: This is the process of expressing a number as the product of its prime factors.

• Long division method: A technique used to find the square root of non-perfect squares through iterative division and subtraction.