Square Root of 118

The square root is the inverse of the square of the number. 118 is not a perfect square. The square root of 118 is expressed in both radical and exponential form. In the radical form, it is expressed as √118, whereas (118)^(1/2) in the exponential form. √118 ≈ 10.86278, 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 A
  
   
• pproximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 118 is broken down into its prime factors.

Step 1: Finding the prime factors of 118 Breaking it down, we get 2 x 59: 2¹ x 59¹

Step 2: Now we found out the prime factors of 118.

Since 118 is not a perfect square, it cannot be grouped into pairs.

Therefore, calculating the square root of 118 using prime factorization is not feasible.

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 118, we need to group it as 18 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 less than or equal to 1. Now the quotient is 1; after subtracting 1 from 1, the remainder is 0.

Step 3: Now let us bring down 18, which is the new dividend. Add the old divisor with the same number 1 + 1, we get 2, which will be 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 ≤ 18. Let us consider n as 4, now 2 x 4 x 4 = 16

Step 6: Subtract 18 from 16; the difference is 2, and the quotient is 14.

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 200.

Step 8: Now we need to find the new divisor that is 8 because 248 x 8 = 1984

Step 9: Subtracting 1984 from 2000, we get the result 16.

Step 10: Now the quotient is 10.8

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

So the square root of √118 ≈ 10.86

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 118 using the approximation method.

Step 1: Now we have to find the closest perfect square of √118. The smallest perfect square less than 118 is 100 and the largest perfect square greater than 118 is 121. √118 falls somewhere between 10 and 11.

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

Going by the formula (118 - 100) / (121 - 100) = 0.8571 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 10 + 0.8571 ≈ 10.86, so the square root of 118 ≈ 10.86

• 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 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. That is the reason it is also known as the principal square root.
   
• Long division method: A method to find the square root of a number by performing division in a step-by-step manner.
   
• Approximation method: A method used to estimate the square root of a number by comparing it with the nearest perfect squares.