Square Root of 156.25

The square root is the inverse of the square of the number. 156.25 is a perfect square. The square root of 156.25 is expressed in both radical and exponential form. In the radical form, it is expressed as √156.25, whereas (156.25)^(1/2) in the exponential form. √156.25 = 12.5, 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 can be used for perfect square numbers, like 156.25. Other methods include the long division method and approximation method. Let us 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 156.25 is broken down into its prime factors.

Step 1: Finding the prime factors of 156.25 Breaking it down, we get 5 x 5 x 5 x 5 x 1.25, which is not required as we have perfect squares. Therefore, we consider 156.25 as a perfect square.

Step 2: Since 156.25 is a perfect square, we can write it as (12.5)^2.

Thus, the square root of 156.25 is 12.5.

The long division method is used for perfect square numbers to ensure accuracy. 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 156.25, we group it as 56.25 and 1.

Step 2: Now we need to find n whose square is 1. We can say n is ‘1’ because 1 x 1 is equal to 1. Now the quotient is 1, and after subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 56.25, which is the new dividend. Add the old divisor with the same number 1 + 1 to get 2, which will be our new divisor.

Step 4: Now, find the number n such that 2n x n ≤ 56.25. For n = 2, 22 x 2 = 44.

Step 5: Subtract 56.25 from 44, the difference is 12.25, and the quotient is 12.

Step 6: 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 1225.

Step 7: Now, find the new divisor, which is 25 because 25 x 5 = 125.

Step 8: Subtracting 125 from 1225, we get the result 0.

Step 9: Now the quotient is 12.5.

So the square root of √156.25 is 12.5.

The approximation method is another method for finding the 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 156.25 using the approximation method.

Step 1: Now we have to find the closest perfect square of √156.25. Since 156.25 is a perfect square, it falls precisely at 12.5.

Step 2: No further approximation is needed, as 156.25 is exactly 12.5.

• Square root: A square root is the inverse of a square. Example: 12.5² = 156.25, and the inverse of the square is the square root, that is, √156.25 = 12.5.
   
• Rational number: A rational number is a number that can be written 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 or a decimal. Example: 156.25 is a perfect square because 12.5^2 = 156.25.
   
• Dividend: The number that is being divided in a division problem. In the context of square roots, it may refer to the number under the square root.
   
• Quotient: The result obtained by dividing one number by another. In the context of square roots, the quotient represents the square root of the dividend.