Square Root of 14096

The square root is the inverse of the square of the number. 14096 is not a perfect square. The square root of 14096 is expressed in both radical and exponential form. In the radical form, it is expressed as √14096, whereas (14096)^(1/2) in the exponential form. √14096 = 118.674, 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:

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 14096 is broken down into its prime factors.

Step 1: Finding the prime factors of 14096. Breaking it down, we get 2 x 2 x 2 x 2 x 13 x 13: 2⁴ x 13².

Step 2: Now we have found the prime factors of 14096. Since 14096 is not a perfect square, calculating it using prime factorization is not straightforward.

However, since the factors can be paired, it shows that 14096 is actually a perfect square.

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, group the numbers from right to left. In the case of 14096, we group it as 96 and 140.

Step 2: The largest number whose square is less than or equal to 140 is 11, because 11 x 11 = 121.

Step 3: Subtract 121 from 140 to get the remainder, which is 19. Bring down the next pair of digits, 96, making it 1996.

Step 4: Double the quotient (11) to get 22, which will be part of the new divisor. Find a digit 'x' such that 22x multiplied by x is less than or equal to 1996.

Step 5: The correct digit 'x' is 8 because 228 x 8 = 1824.

Step 6: Subtract 1824 from 1996 to get the remainder, 172.

Step 7: Since we want more decimal places, add a decimal point and bring down two zeros, making the new dividend 17200.

Step 8: The process continues similarly to find more decimal places. The quotient is approximately 118.674. So the square root of √14096 is approximately 118.674.

The approximation method is another method for finding square roots, and 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 14096 using the approximation method.

Step 1: We have to find the closest perfect square of √14096. The smallest perfect square less than 14096 is 13824, and the largest perfect square greater than 14096 is 14400. √14096 falls somewhere between 118 and 120.

Step 2: Now apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (14096 - 13824) ÷ (14400 - 13824) = 272 ÷ 576 = 0.472.

Using this approximation, add the value to the lower square root: 118 + 0.472 = 118.472. So the approximate square root of 14096 is 118.472.

• Square root: A square root is the inverse operation of squaring a number. Example: 4² = 16, and the inverse of the square is the square root, which is √16 = 4.

• Irrational number: An irrational number is a number that cannot be written in the form p/q, where q is not equal to zero and p and q are integers.

• Perfect square: A perfect square is a number that is the square of an integer. Example: 144 is a perfect square because it is 12².

• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. Example: 14096 can be expressed as 2⁴ x 13².

• Long division method: A method used to find the square root of numbers, especially non-perfect squares, involving division steps to reach an accurate result.