Square Root of 125

The square root of 125 is ±11.1803398875. Finding the square root is just the inverse of squaring a number and hence, squaring 11.1803398875 will result in 125.  The square root of 125 is written as √125 in radical form. In exponential form, it is written as (125)^1/2 

We can find the square root of 125 through various methods. They are:

• Prime factorization method

• Long division method

• Approximation/Estimation method

The prime factorization of 125 is done by dividing 125 by prime numbers and continuing to divide the quotients until they can’t be divided anymore.

 Find the prime factors of 125

After factorizing 125, make pairs out of the factors to get the square root.

 If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.

So, Prime factorization of 125 = 5 × 5 × 5   

But here in case of 125, a pair of factors 5 can be obtained and a single 5 is remaining

So, it can be expressed as  √125 =  5 × √5 = 5√5

5√5 is the simplest radical form of √125

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 125:

 Step 1 : Write the number 125, 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 2 and 5 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 25. Here, that number is 1. 
21×1=21<25.

Step 7: Subtract 25-21=4. Add a decimal point after the new quotient 11, again, bring down two zeroes and make 4 as 400. 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, 7600 (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 11.180….

Follow the steps below:

Step 1: Find the nearest perfect square number to 125. Here, it is 121 and 144.

Step 2: We know that, √121=11 and √144=12. This implies that √125 lies between 11 and 12.

Step 3: Now we need to check √125 is closer to 11 or 12. Let us consider 11 and 11.5. Since (11)²=121 and (11.5)²=132.25. Thus, √125 lies between 11 and 11.5.

Step 4: Again considering precisely, we see that  √125 lies close to (11)2=121. Find squares of (11.1)²=123.21 and (11.3)²= 127.69.

We can iterate the process and check between the squares of 11.15 and 11.2 and so on.

We observe that √125=11.180…

• 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, 2 4 = 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