Square Root of 15129

The square root is the inverse of the square of a number. 15129 is a perfect square. The square root of 15129 is expressed in both radical and exponential forms. In the radical form, it is expressed as √15129, whereas 15129^(1/2) is the exponential form. √15129 = 123, which is a rational number because it can 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. The long division method and approximation method can also be applied. Let us now learn the following methods: -

1. Prime factorization method 
2. Long division method 
3. Approximation method

The product of prime factors is the prime factorization of a number. Here is how 15129 is broken down into its prime factors:

Step 1: Finding the prime factors of 15129. By inspection or trial division, we find that 15129 = 123 × 123. Thus, the prime factorization is 3² × 41²

Step 2: Since 15129 is a perfect square, the digits of the number can be grouped into pairs: (3 × 41) × (3 × 41).

Hence, calculating the square root of 15129 using prime factorization gives us 123.

The long division method can also be used for non-perfect square numbers, but let's apply it to 15129 for illustration:

Step 1: Start by grouping the digits from right to left. For 15129, group as 15 | 129.

Step 2: Find the largest number whose square is less than or equal to 15. This number is 3, because 3 × 3 = 9.

Step 3: Subtract 9 from 15 to get a remainder of 6. Bring down the next pair of digits, 29, to get the new dividend 629.

Step 4: Double the quotient obtained so far (3) and write it as 6_.

Step 5: Find a digit n such that 6n × n is less than or equal to 629. The digit is 3, because 63 × 3 = 189.

Step 6: Subtract 189 from 629 to get 440. Bring down the next pair of zeros to get 4400.

Step 7: Continue the process. The whole number part of the quotient is the square root. Through this method, we find that the square root of 15129 is 123.

The approximation method involves finding the closest perfect squares. Since 15129 is a perfect square, this method is straightforward:

Step 1: Identify the closest perfect squares around 15129.

Step 2: Since 15129 is a perfect square, we already know its square root is exactly 123.

Step 3: If it were not a perfect square, you would use the formula (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) for a more refined approximation.

• Square root: The square root is the inverse of a square. For example, 123² = 15129, and the inverse of the square is the square root, so √15129 = 123. 

• Rational number: A rational number is a number that can be expressed in the form of p/q, where q is not equal to zero and p and q are integers. 

• Perfect square: A perfect square is a number that is the square of an integer. For example, 15129 is a perfect square because its square root is 123. 

• Prime factorization: Prime factorization is the process of decomposing a number into its prime factors. For example, the prime factorization of 15129 is 3² × 41². 

• Long division method: A method used to find the square root of a number by performing division in a structured manner.