Square root of 193

For non-perfect square numbers like 193, which are not perfect squares, there is no square root, which will be a whole number; that will be some decimal value or irrational number.

The number is multiplied by itself to produce the square root of 193. 193 is NOT a perfect square, and thus the square root of 193 is not an exact whole number, the approximate value of √193 = 13.892

The square of a number is represented as x². 

In radical form, it is written as √193 and (193)^½ in exponential form. 
 

Finding the square root of a number is the process when a number multiplied by itself gives another number. We can find the square root of a number by using methods like Prime Factorization, Long Division method, Approximation method, and Subtraction method.
 

We apply this technique by factoring the number into prime factors.

Step 1: Look for the prime factorization of 193

Prime factors of 193 is 1 and 193 itself

Both 1 and 193 are prime numbers.

Now that we have all the factors and their product.

Step 2: Pair them (if possible).

But we cannot pair 193 as identical prime factors. Prime factorization will not help us simplify the square root of 193 in this case due to having no perfect square factors.

This means that prime factorization of the non-perfect-squares like 193 does not give simple factors, since it is already the most simplified form. For numbers that are not perfect squares, we have to use methods like long division or estimation.
 

In the case of non-perfect squares, we can use long division to get a more accurate decimal place.

Step 1: Begin by pairing the digits of 193 from the right. As 193 has three numbers, consider “1” and “93” as a combination.

Step 2: Pick the largest number which, when squared, results in a number less than or equal to (√1) group. That number is 1, as 1×1=1. 

Step 3: Drag down the pair of 93 at the right of the remainder 0, and now we have our new division which is 93. 

Step 4: Now add 1, the last digit of the quotient to the divisor that is 1 so 1+1=2 and now to the right of 2, find a digit such that 2X is less than or equal to 93. When we find X, X is 3, so the new divisor is 23 for the new dividend 93.

Step 5: Now the next thing you need to do is divide 93 by 23 which will result in 3 as our quotient and give us a remainder, 93 - (23×3) = 93–69 = 24

Step 6: Drag down the pair of zeroes to the right of 24, and now we have our new remainder as 2400. 

Step 7: We need to add the last digit of the quotient to the divisor, which is 3+23=26, and need to find a digit such that 26y is less than or equal to 2400. So now together it will form a new divisor that is 268 for the new dividend of 2400. 

Step 8: 2400 is divided by 268, and we get the quotient as 8 which will give us the remainder as 2400 - (268×8) = 2400–2144 = 256

Step 9: Drag down the pair of zeroes and keep repeating the above steps till the desired decimal values. 
From the above calculations and steps, we can conclude that the square root of 191 is ±13.8
 

To find the square root of 193, we start by finding the two perfect squares in which 193 lies. Since 14² =196 and 13² =169, so now we know that the square root of 193 is between 13 and 14.

This process estimates the square root of a number

We start by guessing 13.9 which is nearest to 14.

13.9²= 193.21, which is a slightly larger number than 193, now we move to a lesser number.

13.8²= 190.44, which is too low 

We can next try 13.85² = 192.122 as this number is also slightly;y lesser than 193 we can try a slightly higher number, 

13.87²=192.75 still not close

13.89²=192.93 which can be rounded off approximately as 193.

After many trial and error, we approximately find the square root of 193 to be 13.8924.
 

• Radicals — It is a symbol represented by √. In math, this symbol is used to represent the square root of a number. 

• Repeated subtraction method — In this method, sequential odd numbers are subtracted from the number until the remainder becomes zero. 

• Radicand  — The number that is under the radical symbol is called radicand. So in √5, 5 is called radicand.