The square root of 81 is a number when multiplied by itself will be 81. It is the positive solution of the quadratic equation x^{2}=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 unit^{2}, 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:

Table of Contents

**Important Things:**

**√81 = 9****81**is a perfect square.**√81**is the radical form of square root of**81**.**81**is the exponential form of square root of^{1/2}**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=9^{2}

Taking square root on both sides, we have

√81 = √9^{2}

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

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

= 9^{1 }= 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, √x^{2} = ±x. Thus, we obtain that √81 = √9^{2} = ±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 unit
^{2}