Square Root of 1125

The square root is the inverse of the square of the number. 1125 is not a perfect square. The square root of 1125 is expressed in both radical and exponential form. In the radical form, it is expressed as √1125, whereas (1125)^(1/2) in the exponential form. √1125 ≈ 33.541, 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 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 1125 is broken down into its prime factors.

Step 1: Finding the prime factors of 1125 Breaking it down, we get 3 x 3 x 5 x 5 x 5: 3^2 x 5^3

Step 2: Now we found out the prime factors of 1125. The second step is to make pairs of those prime factors. Since 1125 is not a perfect square, therefore the digits of the number can’t be grouped in perfect pairs. Therefore, calculating 1125 using prime factorization requires taking one factor from each pair and multiplying them together, √1125 = 3 x 5√5 = 15√5.

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 1125, we need to group it as 11 and 25.

Step 2: Now we need to find n whose square is less than or equal to 11. We can say n is '3' because 3 x 3 = 9, which is less than 11. Now the quotient is 3, and after subtracting 11 - 9, the remainder is 2.

Step 3: Now let us bring down 25, which is the new dividend. Add the old divisor with the same number, 3 + 3, we get 6, which will be our new divisor.

Step 4: The new divisor will be 60 + n, where n is the new digit in the quotient. We need to find the value of n.

Step 5: The next step is finding 60n x n ≤ 225. Let us consider n as 3, now 60 x 3 + 3 x 3 = 189.

Step 6: Subtract 225 from 189, the difference is 36, and the quotient is 33.

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 3600.

Step 8: Now we need to find the new divisor that is 669 because 669 x 5 = 3345.

Step 9: Subtracting 3345 from 3600, we get the result 255.

Step 10: Now the quotient is 33.5.

Step 11: 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 √1125 is approximately 33.54.

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 1125 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1125. The smallest perfect square less than 1125 is 1024, and the largest perfect square greater than 1125 is 1156. √1125 falls somewhere between 32 and 34.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula (1125 - 1024) ÷ (1156 - 1024) = 0.7656. Using the formula, we identified the decimal point of our square root approximation. The next step is adding the value we got initially to the decimal number, which is 32 + 0.7656 ≈ 32.77, so the square root of 1125 is approximately 33.54 when using a more precise calculation.

• Square root: A square root is the inverse of a square. For 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 is used more frequently due to its applications in the real world. This is known as the principal square root.

• Prime factorization: The process of breaking down a number into its basic prime number factors.

• Long division method: A method used to divide large numbers and to find square roots of non-perfect squares through a systematic approach.