Square Root of 115

We can find the square root of 115 through various methods. They are:

• Prime factorization method

• Long division method

• Approximation/Estimation method

The prime factorization of 115 is done by dividing 115 by prime numbers and continuing to divide the quotients until they can’t be divided anymore. 

• Find the prime factors of 115

• After factorizing 115, 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 pair.

So, Prime factorization of 115 = 5 × 23   

But here in case of 115, no pairs of factors can be obtained and a single 5 and a single 23 are remaining. So, it can be expressed as  √115, the simplest radical form of √115. 
 

This is a method used for obtaining the square root for non-perfect squares. 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 115:

 Step 1 : Write the number 115, 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 1 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 1. Here, that number is 0. 
20×0=0<1.

Step 7: Subtract 15-0=15. Add a decimal point after the new quotient 10, again, bring down two zeroes and make 15 as 1500. Simultaneously add the unit’s place digit of 20, i.e., 0 with 20. We get here, 20. 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, 17271 (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 10.723….
 

Approximation or estimation of square root is not the exact square root, but it is an estimate.Here, through this method, an approximate value of square root is found by guessing.

Follow the steps below:

Step 1: Find the nearest perfect square number to 115. Here, it is 100 and 121.

Step 2: We know that, √100=10 and √121=11. This implies that √115 lies between 10 and 11.

Step 3: Now we need to check √115 is closer to 10 or 11. Let us consider 10.5 and 11. Since (10.5)²=110.25 and (11)²=121. Thus, √115 lies between 10.5 and 11.

Step 4: Again considering precisely, we see that  √115 lies close to (10.5)²=110.25. Find squares of (10.6)²=112.36 and (10.8)²= 116.64.

We can iterate the process and check between the squares of 10.7 and 10.75 and so on.

We observe that √115=10.723…
 

• 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