Square root of 81

The square root of 81 is a number when multiplied by itself will be 81. It is the positive solution of the quadratic equation x²=81. The square root of 81 is denoted by the symbol √81. In this section, we will learn how to find the value of √81.

Note that if a square has an area of 81 unit², then we use the square root of 81 to find the length of the square. 

The value of the square root of 81 is 9.

If you want to know more about square roots, visit the following pages:   

• Basic concepts of square roots
• Properties of square roots

Important Things:

• √81 = 9
• 81 is a perfect square.
• √81 is the radical form of square root of 81.
• 81^1/2 is the exponential form of square root of 81.
• The square root of 81 is 9.000 in decimal form.
• Square of 81: 81×81 = 6561

It’s time to evaluate the square root of 81.

What is the Square Root of 81?

Note that the value of the square root of 81 is 9. To prove this, we will write 81 as a square of some numbers as 81 is a perfect square.

See that 81 is a product of two numbers of 9‘s. So we write

81=9×9=9²

Taking square root on both sides, we have

√81 = √9²

As square root can be written as power 1/2, we get that

√81 = (9²)^1/2 = 9^2×1/2   as we know that (a^m)ⁿ = a^m×n

= 9¹ = 9

So the quadratic root of 81 is 9.

The positive square root of 81: We know that if x is the square root of a number then -x is also a square root of that number, that is, √x² = ±x. Thus, we obtain that √81 = √9² = ±9. Hence, 9 is the positive square root of 81 and -9 is the negative square root of 81. 

 

Is 81 a perfect square number?

As we know that √81=9 and 9 is a whole number, we have the square root of 81 is also a whole number. Thus the definition of a perfect square implies that 81 is a perfect square number.

 

Square root of 81 by Prime Factorization

We will now find the square root of 81 by the prime factorization method. At first, we have to factorize 81.

As the sum of the digits of 81, that is, 8+1=9 is divisible by 3, we conclude that 81 is divisible by 3. So we write 

81=3×27.

By the above logic, 27 is also divisible by 3 and we have

27=3×9

As 9=3×3 we obtain that

81=3×3×3×3 …(∗)

So the above is the prime factorization of 81.

Taking square root on both sides of (∗), we get that

√81 = √(3×3×3×3)

= √(3×3) × √(3×3)

= 3×3  as we know that √a×a=a

= 9 

∴  the square root of 81 is 9. 

Is Square Root of 81 Rational?

By definition, a rational number can be written as p/q where p and q are integers with q ≠ 0.

We have calculated that √81=9.

As 9=9/1, the number 9 can be written as p/q with p=9, q=1≠0.

∴ 9 is a rational number.

⇒ the square root of 81 is also a rational number.

Conclusion: √81 is not an irrational number.

 

Key Things about 81

• 81 is a composite odd number.
• A square of length 9 unit has the area 81 unit²