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Square Root

What Is the Square Root of 81? How To Find the Square Root of 81?

Written by Prerit Jain

Updated on: 12 Aug 2023

What Is the Square Root of 81? How To Find the Square Root of 81?

What Is the Square Root of 81? How To Find the Square Root of 81?

The square root of 81 is 9. The square root of any number is just the inversion of squaring the given number. The number when multiplied twice gives the original value. For the given number 81, when the number 9 is squared 92 or when multiplied twice by 9*9 we get 81.

For a perfect square number 81 finding the square root of the number is simple as they are whole numbers and not integers. Long division is used to find the square root of irrational and non-perfect square numbers. Since 81 can be the ratio of two numbers it is a rational number p/q, q = 0.

Methods to find the square root of 81

Three methods are used to find the square root of a number that has perfect squares(here 81),

  • Long division method
  • Repeated subtraction method
  • Prime factorization method

Long division method

Finding the divisor for the long division method is much simpler than those that are imperfect squares. When 9 is multiplied twice it gives 81 hence, the number 9 is used as a divisor. 

Hence, the √81 = 9

Repeated subtraction method

Using the repeated subtraction method, the square root of the number can be easily solved but consumes time. Hence, the prime factorization and the long division method are widely used to find the square root of the number.

Start subtracting the given number by the first odd number, take the given value, and further subtract the number with the next odd number repeat the step until you reach zero. 

  1. 81 – 1 = 80
  2. 80 – 3 = 77
  3. 77 – 5 = 72
  4. 72 – 7 = 65
  5. 65 – 9 = 56
  6. 56 -11 = 45
  7. 45 -13 = 32
  8. 32 -15 = 17
  9. 17 – 17 = 0

The last step in which zero is obtained is the 9th step.

Hence, the √81 = 9

Prime factorization method

Prime factorization is one of the simplest methods used to find the square root of a given number. Take the prime factor of the number as n square the given number we get n2, take a single value, and multiply with the other square. By doing so we will get the square root of the number.

  • Step 1: Find the prime factor of 81
    • 81 = 3 × 3 ×  3 × 3
    • 81 = ((3 × 3 × 3 × 3))2
  • Step 2: Find the square of the similar factors by grouping them 

81 = 3 × 3

  • Step 3: By multiplying the common factors we get the square root of the number
    • √81 = 9

Solved examples

Example 1: What is the sum of the square of 81 and the square root of 81?
Square of 81= 6561
The square root of 81 = 9
The sum of the square of 81 and the square root of 81 = 6561+9 = 6570

Example 2: Karen wants to find the square root value of 9/81 to help him solve the sum
The 9 = 3.
The 81 = 9
Therefore, the 9/81 = 1/3.

Example 3: Solve 281 +  281
The 81 = 9.
Hence 2 (9) + 2 (9).
= 18+18
=36.

Example 4: what is the area of the square whose each side is equal to 9cm
Area of the square = x2
Here x= 9
Therefore, 92 = 9*9 = 81
Hence the area of the square with 9cm = 81 sq. cm.

Frequently asked questions

What is the square root of 81?
The square root of 81 = 9.

Explain why the square root of 81 is a rational number.
When finding the prime factor of 81 we get 92 which is entirely an even power and hence all the numbers are positive integers which makes them rational numbers.

What is the value of 9 and the square root of 9?
The square root of 81 is 9, therefore 8181 = 9*81 = 729

What is the real value for the square root of 81?
The actual value for the 81 = 9

What is the square root of 1?
The square root of 1 = 1

What is the square root of (-81)?
The square root of minus digits is imaginary numbers or imaginary units which really don’t have a square root.

Is 81 a perfect square number?
Yes, 81 is a perfect square number because they give the whole numbers when divided

Written by

Prerit Jain

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