Square Root of 8825

The square root is the inverse of the square of the number. 8825 is not a perfect square. The square root of 8825 is expressed in both radical and exponential form. In the radical form, it is expressed as √8825, whereas in the exponential form it is expressed as 8825^(1/2). √8825 ≈ 93.9685, 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, for non-perfect square numbers, the long division method and approximation method are used. Let us now learn about 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 8825 is broken down into its prime factors.

Step 1: Finding the prime factors of 8825

Breaking it down, we get 5 x 5 x 7 x 13 x 19: 5^2 x 7 x 13 x 19

Step 2: Now we have found the prime factors of 8825. The second step is to make pairs of those prime factors. Since 8825 is not a perfect square, the digits of the number can’t be completely grouped in pairs. Therefore, calculating √8825 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 8825, we need to group it as 88 and 25.

Step 2: Now we need to find n whose square is less than or equal to 88. We can say n is ‘9’ because 9 x 9 = 81 which is less than 88. Now the quotient is 9, after subtracting 81 from 88, the remainder is 7.

Step 3: Now let us bring down 25, which is the new dividend. Add the old divisor with the same number (9 + 9) to get 18, which will be our new divisor.

Step 4: The new divisor will be 18_, and we need to find the value of n.

Step 5: The next step is finding 18n x n ≤ 725. Let us consider n as 4. Now, 184 x 4 = 736, which exceeds 725. Try n as 3. Now 183 x 3 = 549 which is less than 725.

Step 6: Subtract 549 from 725, the difference is 176, and the quotient is 93.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 17600.

Step 8: Now we need to find the new divisor, which is 939. We need to find n such that 939n x n is closest to 17600.

Step 9: Continuing this process will give us the approximate value of the square root of 8825.

The approximation method is another method for finding square roots. It is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 8825 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √8825. The smallest perfect square less than 8825 is 8836, and the largest perfect square greater than 8825 is 8712. Therefore, √8825 falls somewhere between 93 and 94.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula, (8825 - 8649) / (8836 - 8649) = 0.9685

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 93 + 0.9685 = 93.9685. Thus, the square root of 8825 is approximately 93.9685.

• Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the square root is √16 = 4.
   
• Irrational number: An irrational number cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.
   
• Long division method: A method used for finding the square roots of non-perfect squares through a systematic approach of division.
   
• Approximation: A method of estimating a value between two known values.
   
• Decimal: A number that includes a whole number and a fractional part, represented with a decimal point, such as 7.86, 8.65, and 9.42.