Square Root of 8704

The square root is the inverse of the square of the number. 8704 is not a perfect square. The square root of 8704 is expressed in both radical and exponential form. In radical form, it is expressed as √8704, whereas (8704)^(1/2) in exponential form. √8704 ≈ 93.273, 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 long-division and approximation methods 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 8704 is broken down into its prime factors.

Step 1: Finding the prime factors of 8704 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 17: 2^9 x 17

Step 2: Now we found out the prime factors of 8704. The second step is to make pairs of those prime factors. Since 8704 is not a perfect square, the digits of the number can’t be grouped in pairs such that all are used.

Therefore, calculating √8704 using prime factorization involves approximating the root of the leftover factor.

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 8704, we need to group it as 87 04.

Step 2: Now, we need to find n whose square is less than or equal to 87. We can say n as ‘9’ because 9 x 9 = 81 is less than or equal to 87. Now the quotient is 9, and after subtracting 81 from 87, the remainder is 6.

Step 3: Now let us bring down 04, which is the new dividend. Add the old divisor with the same number 9 + 9 = 18, which will be our new divisor.

Step 4: 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 × n ≤ 604. Let us consider n as 3, now 183 x 3 = 549.

Step 6: Subtract 549 from 604; the difference is 55, 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 zeroes to the dividend. Now, the new dividend is 5500.

Step 8: Now, we need to find the new divisor that is 932 because 932 x 6 = 5592.

Step 9: Subtracting 5592 from 5500, we get the result -92, indicating the need for a smaller n in the previous step for accuracy.

Step 10: 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 √8704 is approximately 93.27.

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 8704 using the approximation method.

Step 1: Now we have to find the closest perfect square of √8704. The smallest perfect square less than 8704 is 8281 (91²), and the largest perfect square greater than 8704 is 8836 (94²). √8704 falls somewhere between 91 and 94.

Step 2: Now we need to apply the formula that is

(Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square).

Going by the formula (8704 - 8281) ÷ (8836 - 8281) = 423 ÷ 555 ≈ 0.76.

Using the formula, we identified the decimal point of our square root.

The next step is adding the value we got initially to the estimated integer which is 91 + 0.76 = 91.76, so the square root of 8704 is approximately 93.27.

• 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 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 process of expressing a number as the product of its prime factors.
   
• Long division method: A step-by-step approach used to find the square root of a number, especially when dealing with non-perfect squares.