Square Root of 8900

The square root is the inverse of the square of the number. 8900 is not a perfect square. The square root of 8900 is expressed in both radical and exponential form. In the radical form, it is expressed as √8900, whereas (8900)^(1/2) in the exponential form. √8900 ≈ 94.3398, 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:

• 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 8900 is broken down into its prime factors.

Step 1: Finding the prime factors of 8900

Breaking it down, we get 2 x 2 x 5 x 5 x 89: 2^2 x 5^2 x 89

Step 2: Now that we found the prime factors of 8900, the second step is to make pairs of those prime factors. Since 8900 is not a perfect square, the digits of the number can’t be grouped into pairs. Therefore, calculating √8900 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 8900, we need to group it as 89 and 00.

Step 2: Now we need to find n whose square is less than or equal to 89. We can say n is ‘9’ because 9 x 9 = 81 is less than 89. Now the quotient is 9, and after subtracting 81 from 89, the remainder is 8.

Step 3: Now let us bring down 00, which is the new dividend. Add the old divisor with the same number 9 + 9, which equals 18, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 18n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 18n x n ≤ 800. Let us consider n as 4, now 184 x 4 = 736.

Step 6: Subtract 800 from 736; the difference is 64, and the quotient is 94.

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 zeroes to the dividend. Now the new dividend is 6400.

Step 8: Now we need to find the new divisor, which is 943 because 943 x 3 = 2829.

Step 9: Subtracting 2829 from 6400, we get the result 3571.

Step 10: Now the quotient is 94.3

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero.

So the square root of √8900 is approximately 94.34.

The approximation method is another method for finding the 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 8900 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √8900. The smallest perfect square less than 8900 is 8836, and the largest perfect square greater than 8900 is 9025. √8900 falls somewhere between 94 and 95.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula: (8900 - 8836) ÷ (9025 - 8836) = 64 ÷ 189 = 0.338

Using the formula, we identified the decimal portion of our square root. The next step is adding the value we got initially to the decimal number, which is 94 + 0.338 = 94.338, so the square root of 8900 is approximately 94.34.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
   
• Prime factorization: The expression of a number as the product of its prime numbers is called prime factorization.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.