Square root of 48

The square root of 45 is ±6.928. The positive value, 6.928 is the solution of the equation x² = 48. As defined, the square root is just the inverse of squaring a number, so, squaring 6.928 will result in 48.  The square root of 48 is expressed as √48 in radical form, where the ‘√’  sign is called the “radical”  sign. In exponential form, it is written as (48)^1/2  
 

We can find the square root of 48 through various methods. They are:

• Prime factorization method

• Long division method

• Approximation/Estimation method

The prime factorization of 48 involves breaking down a number into its factors. Divide 48 by prime numbers, and continue to divide the quotients until they can’t be separated anymore.

After factoring 48, 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 48 = 2 × 2 ×2 × 2 ×3 

for 48, two pairs of factors 2 can be obtained, and a single 3 is remaining.

So, it can be expressed as  √48 = √(2× 2 ×2  × 2 ×3) = (2× 2)√3=  4√3

4√3 is the simplest radical form of √48.

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 48:

Step 1 : Write the number 48, 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 6, Because 6²=36 < 48

Step 3 : Now divide 48 by 6 (the number we got from Step 2) such that we get 6 as quotient, and we get a remainder. Double the divisor 6, we get 12 and then the largest possible number A1=9 is chosen such that when 9 is written beside the new divisor, 12, a 3-digit number is formed →129 and multiplying 9 with 129 gives 1161 which is less than 1200.

Repeat the process until you reach remainder 0

We are left with the remainder, 2816 (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 6.928

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 : Identify the square roots of the perfect squares above and below 48

Below : 36→ square root of 36 = 6     ……..(i)

Above : 49 →square root of 49 = 7     ……..(ii)

Step 2 : Divide 48 with one of 6 or 7 

 If we choose 7, and divide 48 by 7, we get 6.85714  …….(iii)

             
Step 3: Find the average of 7 (from (ii)) and 6.85714 (from (iii))

(7+6.85714)/2 = 6.928

            
 Hence, 6.928 is the approximate square root of 48
 

• 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