Square Root of 81
81 is a perfect square number and it is the fourth power of 3. i.e. ( 3^{4 }). In this lesson, we will calculate the square root of 81 by repeated subtraction method and solve a few interesting problems.
 Square Root of 81: √81 = 9
 Square of 81: 81² = 6,561
What Is the Square Root of 81?
 The square root of a number is the number that when multiplied to itself gives the original number as the product. Finding the square root of a number is the inverse of squaring a number.
 81 = a × a = a^{2 }
 Thus, a = √81= √(9 × 9)
 9 × 9 = 81 or 9 × 9 = 81. Therefore √81= ± 9
 This shows that 81 is a perfect square.
Is the Square Root of 81 Rational or Irrational?
 A rational number is defined as a number that can be expressed in the form of a quotient or division of two integers, i.e. p/q, where q is not equal to 0.
 9 and 9 can be expressed as 9/1 and 9/1
 Both numbers can be represented in the form of a rational number.
 Hence, the square root of 81 is a rational number.
How to Find the Square Root of 81?
The square root of 81 can be calculated using different methods such as prime factorization or the repeated subtraction method.
Square Root of 81 by Repeated Subtraction Method
Start from 81 and keep subtracting the successive odd numbers till we obtain zero. The total numbers we subtract is the square root of 81.
 81  1 = 80
 80  3 = 77
 77  5 = 72
 72  7 = 65
 65  9 = 56
 56 11 = 45
 45 13 = 32
 32 15 = 17
 17  17 = 0
Thus starting from 81, we have subtracted 9 times to obtain 0. Thus, the square root of 81 is 9.
Square Root of 81 by Prime factorization method
 Prime factorization is expressing the number as a product of its prime factors.
 The prime factor of 81 is 3. 81 = 3 × 3 × 3 × 3
 The square root of 81 is √81 = √( 3 × 3 × 3 × 3)
 √81 = √( 9 × 9)
 (81^{½ })^{2 }= ( 9 ^{½ })^{2}
 Squaring on both the sides, we get 81 = 9
Explore Square roots using illustrations and interactive examples
Important Notes
 The square root of 81 is expressed as √81 in the radical form and as 81^{½} in exponential form.
 The square root of 81 means the second root of 81 = +9 or 9
 The square root of only perfect squares can be calculated easily using the prime factorization method or repeated subtraction method. 81 is a perfect square.
Challenging Questions
 Mike uses a ladder of length 15 feet and starts painting the wall. The foot of the ladder is 9 feet away. If the height of the wall from the ground is 12 feet, then how far is the foot of the ladder from the ground?
Square Root of 81 Solved Examples

Example 1: Sam has decided to plant 81 roses in his garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.
Solution:
Number of rows × number of rose plants = 81 rose plants
Given that the number of rows = the number of plants = n
n × n = 81
n^{2} = 81
n = ± 9
Rows and plants are numbered in positive only.
Therefore, Sam can have 9 rows of 9 rose plants each. 
Example 2: Danio finds a square digital image on a website. The image contained 810,000 pixels. How high is the image in pixels?
Solution:
The pixels occupied by the image = area of the square image = 810,000 pixels.
The length or height of the image = side of the square
side × side = 810000
side = √ 810000
= (81 × 10000)^{½}
= (9^{2} × 100^{2})^{½}
= (9 × 100) = 900 pixels
FAQs on Square Root of 81
What is the square root of 81?
The square root of 81 is +9 or 9.
Why is the square root of 81 negative 9?
81 =  9 x 9. Thus, the square root of 81 can be negative 9.
Is 81 an irrational number?
No, 81 is a rational number. It can be expressed as a rational number as 81/1
Between which two numbers does the square root of 81 lie?
The square root of 81 lies between 8 and 10.
What are the methods to find the square root of 81?
We can find the square root of 81 using prime factorization method, or repeated subtraction method.
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