Last updated on May 26th, 2025
The square root of 72 is a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 72. The number 72 has a unique non-negative square root, called the principal square root.
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.
When we find the square root of 72, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.
If x= √72, what is x2-2 ?
X= √72
⇒ x2 = 72
⇒ x2-2 = 72-2
⇒ x2-2 = 70
Answer : 70
We did the square of the given value of x and then subtracted 2 from it.
Find the length of a side of a square whose area is 72 cm2.
Given, the area = 72 cm2
We know that, (side of a square)2 = area of square
Or, (side of a square)2 = 72
Or, (side of a square)= √72
Or, side of a square = 8.48525. But, the length of a square is a positive quantity only, so, the length of the side is 8.48525 cm.
Answer: 8.48525 cm
We know that, (side of a square)2 = area of square. Here, we are given with the l area of the square, so, we can easily find out its square root because its square root is the measure of the side of the square.
Simplify (√72 + √72) ⤫ √72
(√72 + √72) ⤫ √72
= (8.4852 + 8.4852) ⤫ 8.4852
= 16.9704 ⤫ 8.4852
= 143.997
Answer: 143.997
We first solved the part inside the brackets, i.e., √72 + √72, which resulted into 22.181 and then multiplying it with √72 which is 8.4852 we get 143.997
If y=√72, find y2
Firstly, y=√72= 8.4852
Now, squaring y, we get,
y2= (8.4852)2=72
or, y2=72
Answer : 72
Squaring “y” which is same as squaring the value of √72 resulted to 72
Calculate (√72/8 + √72/9)
√72/8 + √72/9
= 8.4852/ 8 + 8.4852/9
= 1.06065 + 0.9428
= 2.5493
Answer : 2.5493
From the given expression, we first found the value of square root of 72 then
solved by simple divisions and then simple addition.
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
Expressing the given expression as a product of its factors
Ex: 48=2 ⤬ 2 ⤬ 2 ⤬ 2 ⤬ 3
Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....
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 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
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