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Last updated on May 26th, 2025

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Square root of 72

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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.

Square root of 72 for US Students
Professor Greenline from BrightChamps

What Is the 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

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Finding the Square Root of 72

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

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Square Root of 72 By Prime Factorization 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 

 

Professor Greenline from BrightChamps

Square Root of 72 By Long Division Method

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….


 

Professor Greenline from BrightChamps

Square Root of 72 By Approximation

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.

Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in the 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.

Mistake 1

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Misunderstanding symbol

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Often when  √72 is mistaken as 722, we square the number 72 and the get the result as 5184. So, understanding of symbol should be clear

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Square Root of 72 Examples

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Problem 1

If x= √72, what is x2-2 ?

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 X= 72

 ⇒ x2 = 72

 ⇒ x2-2 = 72-2

 ⇒ x2-2 = 70

Answer : 70

 

Explanation

We did the square of the given value of x and then subtracted 2 from it.

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Problem 2

Find the length of a side of a square whose area is 72 cm2.

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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

 

Explanation

 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.

 

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Problem 3

Simplify (√72 + √72) ⤫ √72

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(72 + 72) ⤫ 72

 = (8.4852 + 8.4852) 8.4852

 = 16.9704  8.4852

 = 143.997

Answer: 143.997

 

Explanation

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

 

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Problem 4

If y=√72, find y2

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 Firstly, y=√72= 8.4852

 Now, squaring y, we get, 

 y2= (8.4852)2=72

 or, y2=72

Answer : 72

 

Explanation

Squaring “y” which is same as squaring the value of √72 resulted to 72

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Problem 5

Calculate (√72/8 + √72/9)

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√72/8 + √72/9

8.4852/ 88.4852/9

= 1.06065 + 0.9428

= 2.5493 

 Answer : 2.5493

 

Explanation

From the given expression, we first found the value of square root of 72 then 

solved by simple divisions and then simple addition.

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FAQs on 72 Square Root

1.What are the factors of 72?

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2.What is the cube root of 72?

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3.Is 72 a perfect square or a non-perfect square?

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4.Is the square root of 72 a rational or irrational number?

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5. What is the simplest form of √75?

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6.How does learning Algebra help students in United States make better decisions in daily life?

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7.How can cultural or local activities in United States support learning Algebra topics such as Square root of 72?

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8.How do technology and digital tools in United States support learning Algebra and Square root of 72?

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9.Does learning Algebra support future career opportunities for students in United States?

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Professor Greenline from BrightChamps

Important Glossaries for 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

Professor Greenline from BrightChamps

About BrightChamps in United States

At BrightChamps, we understand algebra is more than just symbols—it’s a gateway to endless possibilities! Our goal is to empower kids throughout the United States to master key math skills, like today’s topic on the Square root of 72, with a special emphasis on understanding square roots—in an engaging, fun, and easy-to-grasp manner. Whether your child is calculating how fast a roller coaster zooms through Disney World, keeping track of scores during a Little League game, or budgeting their allowance for the latest gadgets, mastering algebra boosts their confidence to tackle everyday problems. Our hands-on lessons make learning both accessible and exciting. Since kids in the USA learn in diverse ways, we customize our methods to suit each learner’s style. From the lively streets of New York City to the sunny beaches of California, BrightChamps brings math alive, making it meaningful and enjoyable all across America. Let’s make square roots an exciting part of every child’s math adventure!
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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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