Square Root of 148

The square root is the inverse of the square of a number. 148 is not a perfect square. The square root of 148 is expressed in both radical and exponential form. In the radical form, it is expressed as √148, whereas (148)^(1/2) is the exponential form. √148 ≈ 12.1655, 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, methods like long division and approximation 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 148 is broken down into its prime factors.

Step 1: Finding the prime factors of 148 Breaking it down, we get 2 x 2 x 37: 2² x 37¹

Step 2: Now we found out the prime factors of 148. Since 148 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 148 using prime factorization is not straightforward.

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 148, we need to group it as 48 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 lesser than or equal to 1. Now the quotient is 1; after subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 48, which is the new dividend. Add the old divisor with the same number: 1 + 1, we get 2 as our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 48; let us consider n as 2, now 2 x 2 x 2 = 8

Step 6: Subtract 48 from 8; the difference is 40, and the quotient is 12.

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 4000.

Step 8: Now we need to find the new divisor that is 91 because 241 x 9 = 2169

Step 9: Subtracting 2169 from 4000, we get the result 1831.

Step 10: Now the quotient is 12.1

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

So the square root of √148 is approximately 12.16.

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 148 using the approximation method.

Step 1: Now we have to find the closest perfect square of √148. The smallest perfect square less than 148 is 144, and the largest perfect square greater than 148 is 169. √148 falls somewhere between 12 and 13.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Going by the formula (148 - 144) / (169 - 144) = 0.16 Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 12 + 0.16 = 12.16.

So the square root of 148 is approximately 12.16.

• Square root: A square root is the inverse of a square. 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 both 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, it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.
   
• Prime factorization: This is the process of breaking down a number into its prime factors. For example, the prime factorization of 148 is 2 x 2 x 37.