Square Root of 8725

The square root is the inverse of the square of the number. 8725 is not a perfect square. The square root of 8725 is expressed in both radical and exponential form. In the radical form, it is expressed as √8725, whereas (8725)^(1/2) in the exponential form. √8725 ≈ 93.3516, 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 8725 is broken down into its prime factors:

Step 1: Finding the prime factors of 8725.

Breaking it down, we get 5 × 5 × 349: 5^2 × 349.

Step 2: Now we found out the prime factors of 8725. The second step is to make pairs of those prime factors. Since 8725 is not a perfect square, it is impossible to pair all digits. Therefore, calculating √8725 using prime factorization is not feasible.

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 8725, we need to group it as 87 and 25.

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

Step 3: Bring down 25, making the new dividend 625. Add the old divisor (9) to itself to get 18 as part of the new divisor.

Step 4: Determine n such that 18n × n ≤ 625. Let n be 3, then 183 × 3 = 549.

Step 5: Subtract 549 from 625, resulting in a remainder of 76. The quotient so far is 93.

Step 6: Since the remainder is still there, add a decimal point to the quotient and bring down 00, making the new dividend 7600.

Step 7: Determine the next digit for the quotient. The new divisor becomes 1860. Choose n such that 1860n × n ≤ 7600.

Step 8: After determining n and performing the steps, continue until the desired precision is achieved.

So the square root of √8725 is approximately 93.35.

The approximation method is another method for finding 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 8725 using the approximation method.

Step 1: Find the closest perfect squares around 8725. The nearest perfect squares are 8464 (92^2) and 8836 (94^2). √8725 falls between 92 and 94.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (8725 - 8464) / (8836 - 8464) = 261 / 372 = 0.701. Add the decimal 0.701 to the lower square root estimate, 92 + 0.701 = 92.701. Rounding gives us approximately 93.35, so √8725 ≈ 93.35.

• 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, 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, it is usually the positive square root that is considered more prominently due to its uses in the real world. This is known as the principal square root.
   
• 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.
   
• Approximation: The process of finding a value that is close enough to the right answer, usually within an acceptable error range.