Square root of 72

The square root of 72 is ±8.4852, where 8.4852 is the positive solution of the equation x² = 72. Finding the square root is just the inverse of squaring a number and hence, squaring 8.4852 will result in 72.  The square root of 72 is written as √72 in radical form, where the ‘√’  sign is called the “radical”  sign. In exponential form, it is written as (72)^1/2

We can find the square root of 72 through various methods. They are:

i) Prime factorization method

ii) Long division method

iii) Approximation/Estimation method

The prime factorization of 72 is done by dividing 72 by prime numbers and continuing to divide the quotients until they can’t be separated anymore. After factorizing 72, 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 72 = 2 × 2 × 2 × 3 × 3  

But here in the case of 72, a pair of factor 2 and factor 3 can be obtained and a single 2 is remaining

So, it can be expressed as  √72 =  2 × 3 × √2 = 6√2

 6√2 is the simplest radical form of √72 

This method is 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.

Follow the steps to calculate the square root of 72:

 Step 1: Write the number 72, 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

               8, Because 8²=64 < 72

Step 3: Now divide 72 by 8 (the number we got from Step 2) such that we get 8 as the

            quotient and we get a remainder. 

             Double the divisor 8, we get 16 and then the largest possible number A1=4 is

             chosen such that when 4 is written beside the new divisor, 16, a 3-digit number is

             formed →164 and multiplying 4 with 164 gives 656 which is less than 800.

             Repeat the process until you reach the remainder 0

              We are left with the remainder, 4775 (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 8.485….

Approximation or estimation of the square root is not the exact square root, but it is an estimate.

Here, through this method, an approximate value of the square root is found by guessing.

Follow the steps below:

Step 1: identify the square roots of the perfect squares above and below 72

             Below : 64→ square root of 64 = 8     ……..(i)

             Above : 81 →square root of 81 = 9     ……..(ii)

Step 2: Dividing 72 with one of 8 or 9 

            If we choose 8 

            We get 9 when 72 is divided by 8    …….(iii)

Step 3: find the average of 8 (from (i)) and 9 (from (iii))

            (8+9)/2 = 8.5 

 Hence, 8.5 is the approximate square root of 72.

• 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 

• 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