Square Root of 12.5

The square root is the inverse of the square of the number. 12.5 is not a perfect square. The square root of 12.5 is expressed in both radical and exponential form. In the radical form, it is expressed as √12.5, whereas (12.5)^(1/2) in the exponential form. √12.5 ≈ 3.5355339, 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. 12.5 is not an integer, so it cannot be directly factored using prime factorization. Thus, the prime factorization method is not applicable for non-integers like 12.5.

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, consider the number 12.5. Move the decimal two places to the right, making it 1250.

Step 2: Group the digits into pairs starting from the decimal point. Consider 12 and 50.

Step 3: Find a number whose square is less than or equal to 12. The number is 3, since 3 × 3 = 9.

Step 4: Subtract 9 from 12, giving 3. Bring down the next pair, 50, to make it 350.

Step 5: Double the divisor (3) to get 6, and determine the next digit of the quotient as 5, because 65 × 5 = 325.

Step 6: Subtract 325 from 350, giving a remainder of 25.

Step 7: Bring down two zeros to make it 2500.

Step 8: The new divisor is 705. Determine the next digit of the quotient. Continue until the desired precision is reached.

So, the square root of 12.5 ≈ 3.5355.

Approximation method is another method for finding square roots. It is an easy way to find the square root of a given number. Let us learn how to find the square root of 12.5 using the approximation method.

Step 1: Find the closest perfect squares to 12.5. The smallest perfect square is 9, and the largest perfect square is 16. √12.5 falls between 3 and 4.

Step 2: Apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). (12.5 - 9) / (16 - 9) = 3.5 / 7 = 0.5

Step 3: Add this value to the smaller number, which is 3.

Therefore, 3 + 0.5 = 3.5, which is an approximation of the square root of 12.5.

• Square root: A square root is the inverse of a square. For example, 4² = 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 always the positive square root that has more prominence due to its uses in the real world. This is why it is also 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.
   
• Long division method: A method used to find the square root of non-perfect square numbers, involving a series of division steps to approximate the value.