Square Root of 137

The square root of 137 is ±11.7046999107.  Basically, finding the square root is just the inverse of squaring a number and hence, squaring 11.7046999107 will result in 137.  The square root of 137 is written as √137 in radical form. In exponential form, it is written as (137)^1/2 
 

We can find the square root of 137 through various methods. They are:

• Prime factorization method

• Long division method

• Approximation/Estimation method
   

The prime factorization of 137 is done by dividing 137 by prime numbers and continuing to divide the quotients until they can’t be divided anymore. 

• Find the prime factors of 137→ The number 137 is a prime number, therefore its factors are only 1 and 137.

• After factorizing 137, make pairs out of the factors to get the square root.
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•  If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.
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So, Prime factorization of 137 = 137 × 1    

here in case of 137, no pairs of factors can be obtained but a single 137 is remaining

So, it can be expressed as  √137 =  √(137 × 1) =  √137, the simplest radical form of √137. 
 

This is a method used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.

Follow the steps to calculate the square root of 137:

 Step 1 : Write the number 137, and draw a horizontal bar above the pair of digits from right to left.

Step 2 : Now, find the greatest number whose square is less than or equal to 1. Here, it is 1, Because 1²=1 < 1.

Step 3 : Now divide 1 by 1 such that we get 1 as quotient and then multiply the divisor with the quotient, we get 1

Step 4: Subtract 1 from 1. Bring down 3 and 7 and place it beside the difference 0.

Step 5: Add 1 to same divisor, 1. We get 2.

Step 6: Now choose a number such that when placed at the end of 2, a 2-digit number will be formed. Multiply that particular number by the resultant number to get a number less than 37. Here, that number is 1. 

21×1=21<37.

Step 7: Subtract 37-21=16. Add a decimal point after the new quotient 11, again, bring down two zeroes and make 16 as 1600. Simultaneously add the unit’s place digit of 21, i.e., 1 with 21. We get here, 22. Apply Step 5 again and again until you reach 0. 

We will show two places of precision here, and so, we are left with the remainder, 16384 (refer to the picture), after some iterations and keeping the division till here, at this point 

             
Step 8 : The quotient obtained is the square root. In this case, it is 11.704….

Approximation or estimation of square root is not finding the exact square root, but it is an estimate.Here, through this method, an estimation of the approximate value of square root is what we will do.

Follow the steps below:

Step 1: Find the nearest perfect square number to 137. Here, it is 121 and 144.

Step 2: We know that, √121=11 and √144=12. This implies that √137 lies between 11 and 12.

Step 3: Now we need to check √137 is closer to 11 or 12. Let us consider 11.5 and 12. Since (11.5)²=132.25 and (12)²=144. Thus, √137 lies between 11.5 and 12.

Step 4: Again considering precisely, we see that  √137 lies close to (12)²=144. Find squares of (11.6)²=134.56 and (11.8)²= 139.24.

We can iterate the process and check between the squares of 11.65 and 11.85 and so on.

We observe that √137=11.704…
 

• Exponential form:  An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent. Ex: 2 × 2 × 2 × 2 = 16 Or, 2⁴ = 16, where 2 is the base, 4 is the exponent 

• Prime Factorization:  Expressing the given expression as a product of its factors. Ex: 48=2 × 2 × 2 × 2 × 3

• Prime Numbers: Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....

• Rational numbers and Irrational numbers: The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers. Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. 

• Perfect and non-perfect square numbers:Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :3, 8, 24