Square Root of 8450

The square root is the inverse of the square of the number. 8450 is not a perfect square. The square root of 8450 is expressed in both radical and exponential form. In the radical form, it is expressed as √8450, whereas (8450)^(1/2) in the exponential form. √8450 ≈ 91.9239, 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 8450 is broken down into its prime factors.

Step 1: Finding the prime factors of 8450 Breaking it down, we get 2 x 5 x 5 x 13 x 13: 2^1 x 5^2 x 13^2

Step 2: Now that we have found the prime factors of 8450, the second step is to make pairs of those prime factors. Since 8450 is not a perfect square, the digits of the number can’t be grouped completely into pairs. Therefore, calculating √8450 using prime factorization yields an approximate value.

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 8450, we need to group it as 50 and 84. Step 2: Now we need to find n whose square is closest to 84 without exceeding it. We can use n as ‘9’ because 9 x 9 = 81, which is less than 84. Now the quotient is 9, and after subtracting 81 from 84, the remainder is 3. Step 3: Now let us bring down 50, which is the new dividend. Add the old divisor with the same number: 9 + 9 = 18, which will be our new divisor. Step 4: The new divisor will be 18n. We need to find the value of n such that 18n x n <= 350. Let us consider n as 1. Now, 181 x 1 = 181. Step 5: Subtract 181 from 350, and the difference is 169. Step 6: 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 16900. Step 7: Now we need to find the new divisor that is 91 because 1891 x 9 = 17019. Step 8: Subtracting 17019 from 16900 gives us a result of -119, which indicates a small overestimate, so we adjust accordingly. Step 9: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero. So the square root of √8450 is approximately 91.92.

The approximation method is another method for finding square roots; 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 8450 using the approximation method.

Step 1: We have to find the closest perfect squares to √8450. The smallest perfect square less than 8450 is 8281, and the largest perfect square less than 8450 is 8836. √8450 falls somewhere between 91 and 94.

Step 2: Now we need to apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula (8450 - 8281) / (8836 - 8281) ≈ 0.642, 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 91 + 0.92 = 91.92, so the square root of 8450 is approximately 91.92.

• Square root: A square root is the inverse of a square. 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 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 in real-world applications. This is known as the principal square root.
   
• Prime factorization: This is the process of expressing a number as the product of its prime factors, which is useful in finding square roots.
   
• Long division method: A method used to find the square root of a non-perfect square by dividing the number into groups and calculating step by step to achieve an approximate value.