Square Root of 156

The square root is the inverse of the square of a number. 156 is not a perfect square. The square root of 156 is expressed in both radical and exponential form. In the radical form, it is expressed as √156, whereas in exponential form it is (156)^(1/2). √156 ≈ 12.48999, 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 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 156 is broken down into its prime factors:

Step 1: Finding the prime factors of 156 Breaking it down, we get 2 x 2 x 3 x 13: 2² x 3¹ x 13¹

Step 2: Now we have found the prime factors of 156. The second step is to make pairs of those prime factors. Since 156 is not a perfect square, the digits of the number can’t be grouped in pairs that fully satisfy the square root.

Therefore, calculating √156 using prime factorization directly is not feasible.

The long division method is particularly used for non-perfect square numbers. In this method, we 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, group the numbers from right to left. In the case of 156, we need to group it as 56 and 1.

Step 2: Find a number whose square is less than or equal to 1. This number is 1, as 1 x 1 = 1, which is less than or equal to 1. The quotient is 1, and after subtracting, the remainder is 0.

Step 3: Bring down 56, which is the new dividend. Add the old divisor with the same number: 1 + 1 = 2, which will be our new divisor.

Step 4: 2n is the new divisor, and we need to find the value of n.

Step 5: Find 2n × n ≤ 56. Let us consider n as 2, so 2 x 2 x 2 = 8.

Step 6: Subtract 8 from 56. The difference is 48, and the quotient is 12.

Step 7: Since the dividend is less than the divisor, we add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now, the new dividend is 4800.

Step 8: Find the new divisor. It is 249 because 249 x 9 = 2241.

Step 9: Subtract 2241 from 2400 to get the result 159.

Step 10: The quotient is approximately 12.48.

Step 11: Continue these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √156 is approximately 12.49.

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 156 using the approximation method.

Step 1: Find the closest perfect squares of √156. The smallest perfect square less than 156 is 144, and the largest perfect square more is 169. √156 falls somewhere between 12 and 13.

Step 2: Apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) Using the formula: (156 - 144) / (169 - 144) = 0.48 Add the value obtained to the integer part: 12 + 0.48 = 12.48.

So the square root of 156 is approximately 12.49.

• Square root: A square root is the inverse of a number squared. Example: 4² = 16, and the inverse of squaring is the square root, which is √16 = 4.
   
• 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, but the positive square root is often used in practical applications. This is why it is known as the principal square root.
   
• Approximation: The process of finding a value that is close to the actual value. For example, √156 is approximately 12.49.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing and averaging.