81 is a perfect square number and it is the fourth power of 3. i.e. ( 34 ). In this lesson, we will calculate the square root of 81 by repeated subtraction method and solve a few interesting problems.

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Square Root of 81: √81 = 9Square of 81: 81² = 6,561
1.What Is the Square Root of 81?
2.Is Square Root of 81 Rational or Irrational?
3.How to Find the Square Root of 81?
4.Important Notes on Square Root of 81
5.Challenging Questions
6.FAQs on Square Root of 81

81 = a × a = a2 Thus, a = √81= √(9 × 9)9 × 9 = 81 or -9 × -9 = 81. Therefore √81= ± 9This shows that 81 is a perfect square.

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9 and -9 can be expressed as 9/1 and -9/1Both numbers can be represented in the form of a rational number.Hence, the square root of 81 is a rational number.

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 = 8080 - 3 = 7777 - 5 = 7272 - 7 = 6565 - 9 = 5656 -11 = 4545 -13 = 3232 -15 = 1717 - 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 × 3The square root of 81 is √81 = √( 3 × 3 × 3 × 3)√81 = √( 9 × 9)(81½ )2 = ( 9 ½ )2Squaring on both the sides, we get 81 = 9

Explore Square roots using illustrations and interactive examples


The square root of 81 is expressed as √81 in the radical form and as 81½ in exponential form.The square root of 81 means the second root of 81 = +9 or -9The square root of only perfect squares can be calculated easily using the prime factorization method or repeated subtraction method. 81 is a perfect square.
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?

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 plantsGiven that the number of rows = the number of plants = nn × n = 81 n2 = 81n = ± 9 Rows and plants are numbered in positive only.Therefore, Sam can have 9 rows of 9 rose plants each.

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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 squareside × side = 810000side = √ 810000= (81 × 10000)½= (92 × 1002)½= (9 × 100) = 900 pixels


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