Square Root of 15

The square root of 15 is ±3.87298334621, where 3.87298334621 is the positive solution of the equation x² = 15.  Basically, finding the square root is just the inverse of squaring a number and hence, squaring 3.87298334621 will result in 15.  The square root of 15 is written as √15 in radical form, where the ‘√’  sign is called “radical”  sign. In exponential form, it is written as (15)^1/2 

We can find the square root of 15 through various methods. They are:

• Prime factorization method

• Long division method

• Approximation/Estimation method

The prime factorization of 15 is done by dividing 15 by prime numbers and continuing to divide the quotients until they can’t be divided anymore. After factorizing 15, make pairs out of the factors to get the square root. If there exists numbers which cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs

So, Prime factorization of 15 = 5 × 3   

But here in case of 15, no pair of factors can be obtained but a single 3 and a single 5 are obtained

So, it can be expressed as  √15 =   √(5 × 3) = √15

√15 is the simplest radical form of √15

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 15:

 Step 1 : Write the number 15, and draw a 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. Here, it is 3, Because 3²=9 < 15

Step 3 : Now divide 15 by 3 (the number we got from step 2) such that we get 3 as quotient and we get a remainder. Double the divisor 3, we get 6 and then the largest possible number A1=8 is chosen such that when 8 is written beside the new divisor, 6, a 2-digit number is formed →68 and multiplying 8 with 68 gives 544 which is less than 600.

Repeat the process until you reach remainder 0

We are left with the remainder, 7616 (refer to the picture), after some iterations and keeping the division till here, at this point.

             
Step 4 : The quotient obtained is the square root. In this case, it is 3.872….

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 and identify the square roots of the perfect squares above and below 15

             Below : 9→ square root of 9 = 3     ……..(i)

             Above : 16 →square root of 16 = 4     ……..(ii)

Step 2 : Dividing 15 with one of 3 or 4 
              
             If we choose 4 

            We get 3.75 when 15 is divided by 4    …….(iii)

             
Step 3:   find the average of 4 (from (ii)) and 3.75 (from (iii))

            (4+3.75)/2 = 3.875 

            
 Hence, 3.875 is the approximate square root of 15

• 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