Square Root of 81
2026-02-28 00:45 Diff
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Last updated on August 5, 2025

The square root of 81 is a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 81. The number 81 has a unique non-negative square root, called the principal square root. Square root concept are applied in real life in the field of engineering, GPS and distance calculations, for scaling objects proportionally, etc.

What Is the Square Root of 81?

The square root of 81 is ±9, where 9 is the positive solution of the equation x2 = 81. Finding the square root is just the inverse of squaring a number and hence, squaring 9 will result in 81. The square root of 81 is written as √81 in radical form, where the ‘√’  sign is called the “radical” sign. In exponential form, it is written as (81)1/2 

Finding the Square Root of 81

Square Root of 81 By Prime Factorization Method

The prime factorization of 81 can be found by dividing the number by prime numbers and continuing to divide the quotients until they can’t be separated anymore.


Steps for Prime Factorization of 81:

Step 1:  Find the prime factors of 81.

Step 2: After factorizing 81, make pairs out of the factors to get the square root.

Step 3: If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.

 Step 4:If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.

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Square Root of 81 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 too sometimes.


Follow the steps to calculate the square root of 81:


  Step 1: Write the number 81 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 81. Here, it is 9 because 92=81 

Step 3: now divide 81 by 9 (the number we got from Step 2) such that we get 9 as a quotient, and we get a remainder 0. 

  Step 4: The quotient obtained is the square root of 81. In this case, it is 9.

Square Root of 81 By Subtraction Method


We know that the sum of the first n odd numbers is n2. We will use this fact to find square roots through the repeated subtraction method. Furthermore, we just have to subtract consecutive odd numbers from the given number, starting from 1. The square root of the given number will be the count of the number of steps required to obtain 0. Here are the steps:


Step 1: take the number 81 and then subtract the first odd number from it. Here, in this case, it is 81-1=80


Step 2: we have to subtract the next odd number from the obtained number until it comes zero as a result. Now take the obtained number (from Step 1), i.e., 80, and again subtract the next odd number after 1, which is 3, → 80-3=77. Like this, we have to proceed further.


Step 3: now we have to count the number of subtraction steps it takes to yield 0 finally.  Here, in this case, it takes 9 steps 


So, the square root is equal to the count, i.e., the square root of 81 is ±9.

Common Mistakes and How to Avoid Them in the Square Root of 81

When we find the square root of 81, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.
 

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Problem 1

Find √(121×100×81×64) ?

Okay, lets begin

 √(121×100×81×64)

= 11 ×10×9×8

= 7920


Answer : 7920
 

Explanation

firstly, we found the values of the square roots of 11,10,9 and 8, then multiplied the values.
 

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Problem 2

What is √81 subtracted from √100 and then multiplied by 10 ?

Okay, lets begin

√100 - √81

= 10–9

= 1


Now, 1⤬ 10

= 10 


Answer: 10 
 

Explanation

finding the value of √100 and√81 to find their difference and multiplying the difference by 10.
 

Well explained 👍

Problem 3

Find the radius of a circle whose area is 81π cm².

Okay, lets begin

Given, the area of the circle = 81π cm2


Now, area = πr2 (r is the radius of the circle)


So, πr2 = 81π cm2

We get, r2 = 81 cm2


r = √81 cm


Putting the value of √81 in the above equation, 


We get, r = ±9 cm


Here we will consider the positive value of 9.


Therefore, the radius of the circle is 9 cm.


Answer: 9 cm.
 

Explanation

We know that, area of a circle = πr2 (r is the radius of the circle).According to this equation, we are getting the value of “r” as 9 cm by finding the value of the square root of 81.
 

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Problem 4

Find the length of a side of a square whose area is 81 cm²

Okay, lets begin

Given, the area = 81 cm2


We know that, (side of a square)2 = area of square


Or,  (side of a square)2 = 81


Or,  (side of a square)= √81


Or, the side of a square = ± 9.


But, the length of a square is a positive quantity only, so, the length of the side is 9 cm.


Answer: 9 cm
 

Explanation

We know that, (side of a square)2 = area of square. Here, we are given with the 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
 

Well explained 👍

Problem 5

Find (√81 / √49) /(√49 / √81)

Okay, lets begin

(√81 / √49) /(√49 / √81)

= 81/49  

= 1.653


Answer : 1.653
 

Explanation

 We found out the values of √81×√81 and √49×√49 after simplifying and then divided the values .
 

Well explained 👍

FAQs on Square Root of 81

1.What is a square of 81?

 The square value of 81 is 6561.
 

2.Does 81 have two square roots?

 Yes, 81 have two square roots → +9 and -9.
 

3.Is 81 a perfect square or non-perfect square?

 81 is a perfect square, since 81 =(9)2.
 

4.Is the square root of 81 a rational or irrational number?

The square root of 81 is ±9. So, 81 is a rational number, since it can be obtained by dividing two integers and can be written in the form p/q, where p and q are integers.
 

5.Is 81 a perfect cube root?

No, 81 is not a perfect cube root because on prime factorization of 81 we get 3 ×3 ×3 ×3, where the 3s cannot be grouped as a triplet.

Important Glossaries for Square Root of 81

  • 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, 24 = 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, 2

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