Square Root of 896

The square root is the inverse of squaring a number. 896 is not a perfect square. The square root of 896 is expressed in both radical and exponential form. In the radical form, it is expressed as √896, whereas in exponential form, it is (896)^(1/2). √896 ≈ 29.9333, which is an irrational number because it cannot be expressed as p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, the long division method and the 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 896 is broken down into its prime factors.

Step 1: Finding the prime factors of 896 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 7 = 2^6 x 7

Step 2: Now we have found the prime factors of 896. The next step is to make pairs of those prime factors. Since 896 is not a perfect square, the digits of the number can’t be grouped in pairs completely.

Therefore, calculating √896 using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. In this method, we find 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 896, we group it as 96 and 8.

Step 2: Now we need to find a number whose square is ≤ 8. We can say this number is 2 because 2 x 2 = 4 and 4 ≤ 8. Now the quotient is 2, and after subtracting 4 from 8, the remainder is 4.

Step 3: Bring down 96, making the new dividend 496. Add the old divisor with the same number (2 + 2) to get 4, which will be part of our new divisor.

Step 4: Find a number n such that 4n × n ≤ 496. Considering n as 9, we get 49 x 9 = 441.

Step 5: Subtract 441 from 496, getting the remainder of 55.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeros, making the new dividend 5500.

Step 7: The next step is finding a new divisor. Let’s assume n is 9, giving us 598 x 9 = 5382.

Step 8: Subtract 5382 from 5500 to get a remainder of 118.

Step 9: The quotient now is 29.9.

Step 10: Continue these steps until you get two numbers after the decimal point.

Thus, the square root of 896 is approximately 29.93.

The approximation method is another way to find square roots. Now let us learn how to find the square root of 896 using this method.

Step 1: Find the closest perfect squares around 896.

The smallest perfect square is 841, and the largest perfect square is 900.

Therefore, √896 falls between 29 and 30.

Step 2: Now apply the formula:

(Given number - smallest perfect square) / (Largest perfect square - smallest perfect square).

Using the formula: (896 - 841) / (900 - 841) = 55 / 59 ≈ 0.9322

Adding 29 to 0.9322 gives 29.9322, so the approximate square root of 896 is 29.93.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: 5^2 = 25, so √25 = 5.
   
• Irrational number: An irrational number cannot be written as a simple fraction; it has non-repeating, non-terminating decimals.
   
• Principal square root: The principal square root is the non-negative root of a number, typically used in calculations.
   
• Perfect square: A perfect square is an integer that is the square of an integer. Example: 16 is a perfect square because it is 4^2.
   
• Radical: A symbol (√) used to denote the square root or nth root of a number.