Square Root of 190

The square root is the inverse of the square of the number. 190 is not a perfect square. The square root of 190 is expressed in both radical and exponential form. In the radical form, it is expressed as √190, whereas (190)^(1/2) in exponential form. √190 ≈ 13.784, 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. Now let us look at how 190 is broken down into its prime factors.

Step 1: Finding the prime factors of 190 Breaking it down, we get 2 × 5 × 19.

Step 2: Now we found the prime factors of 190. The second step is to make pairs of those prime factors. Since 190 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 190 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 190, we need to group it as 90 and 1.

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

Step 3: Now let us bring down 90, 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: 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 × n ≤ 90. Let us consider n as 4, now 24 × 4 = 96, which is more than 90. Let us try n as 3, now 23 × 3 = 69.

Step 6: Subtract 90 from 69, the difference is 21, and the quotient is 13.

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 2100.

Step 8: Now we need to find the new divisor, and the next number in the quotient. Let n be 8, then 278 × 8 = 2224.

Step 9: Subtracting 2224 from 2100 results in a negative number, which means 8 is too high. Let's try n as 7, now 277 × 7 = 1939.

Step 10: Subtracting 1939 from 2100 gives 161.

Step 11: Now the quotient so far is 13.7. Continue doing these steps until you 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 √190 is approximately 13.78.

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 190 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √190. The smallest perfect square less than 190 is 169 (13^2), and the largest perfect square greater than 190 is 196 (14^2). √190 falls somewhere between 13 and 14.

Step 2: Now we need to apply the formula that is: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) (190 - 169) / (196 - 169) ≈ 0.78 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 13 + 0.78 = 13.78, so the square root of 190 is approximately 13.78.

• 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.

• Approximation: A method of finding an estimated value that is close to the actual value.

• Long Division Method: A technique used to find the square root of a number by dividing and estimating.