Square Root of 10025

The square root is the inverse of the square of the number. 10025 is not a perfect square. The square root of 10025 is expressed in both radical and exponential form. In radical form, it is expressed as √10025, whereas (10025)^(1/2) in exponential form. √10025 ≈ 100.1249, 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 prime factorization method is not used; instead, 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 10025 is broken down into its prime factors.

Step 1: Finding the prime factors of 10025

Breaking it down, we get 5 x 5 x 401.

Step 2: Now we found out the prime factors of 10025. The second step is to make pairs of those prime factors. Since 10025 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 10025 using prime factorization alone is not straightforward, and approximation or other methods are more useful.

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, group the numbers from right to left. In the case of 10025, group it as 25 and 100.

Step 2: Find a number whose square is less than or equal to 100. The number is 10 because 10 x 10 = 100. Now, the quotient is 10, and the remainder is 0.

Step 3: Bring down 25, which is the new dividend. Add the old divisor with the same number, 10 + 10, to get 20 as the new divisor.

Step 4: Find n such that 20n x n ≤ 25. Let n be 0, so 20 x 0 = 0.

Step 5: Subtract 0 from 25, the difference is 25, and the quotient is 100.

Step 6: Since the dividend is less than the divisor, add a decimal point. Adding the decimal allows us to add two zeros to the dividend. Now the dividend is 2500.

Step 7: Find the new divisor that is 2002 because 2002 x 1 = 2002.

Step 8: Subtract 2002 from 2500 to get 498.

Step 9: Continue these steps until the desired accuracy is achieved.

Eventually, the square root of 10025 ≈ 100.1249.

The approximation method is another method for finding square roots, and it is an easy method for estimating the square root of a given number. Now let us learn how to find the square root of 10025 using the approximation method.

Step 1: Find the closest perfect squares to √10025. The smallest perfect square less than 10025 is 10000 (100^2), and the largest perfect square greater than 10025 is 10201 (101^2). √10025 falls somewhere between 100 and 101.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (10025 - 10000) / (10201 - 10000) = 25 / 201 ≈ 0.1249

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 100 + 0.1249 = 100.1249. So the square root of 10025 is approximately 100.1249.

• Square root: A square root is the inverse of a square. For example, 10^2 = 100, and the inverse of the square is the square root, that is, √100 = 10.
   
• Irrational number: An irrational number 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, known as the principal square root, is often used due to its practical applications.
   
• Factors: Factors are numbers that divide another number completely without leaving a remainder.
   
• Approximation: Approximation involves estimating the value of a number or expression based on close or nearby values.