Square & Square Root of 81

Last Updated: April 28, 2024

Square & Square Root of 81

Square of 81

81² (81×81) = 6561

To calculate the square of 81, you simply multiply 81 by itself:

Therefore, the square of 81 is 6561. This straightforward calculation is a building block for more complex mathematical operations and concepts, including algebraic equations, geometric formulas, and statistical models.

Square Root of 81

√81​ = 9

The square root of 81, denoted as √81, equals 9. To calculate it, you find the number that, when multiplied by itself, yields 81. In mathematical terms, finding a square root means identifying the number that, when raised to the power of 2, equals the original number. Visually, this can be represented as a square with sides of length 9 units, where the area of the square equals 81 square units. Understanding square roots is crucial across various mathematical concepts and applications, such as in geometry, algebra, and in solving problems that involve extracting the original quantity from squared measurements. In practical situations, knowing the square root of 81 is useful for tasks that require reversing the squaring process to find original values.

Square Root of 81: “9”

Exponential Form: 81^1/2 or 81^0.5

Radical Form: 81

Is the Square Root of 81 Rational or Irrational?

The square root of 81 is rational

This is because it can be exactly expressed as 9, which is a quotient of two integers (9/1), fitting the definition of a rational number. Rational numbers are those that can be expressed as a fraction a/b​, where a and b are integers, and b is not zero.

Methods to Find Value of Root 81

Finding the value of the square root of 81 (√81) can be achieved through several methods, each suitable for different contexts:

1. Mental Calculation

Given that 81 is a perfect square, one can determine its square root mentally by recognizing that 9×9=81. Thus, √81 = 9.

2. Prime Factorization

Break down 81 into its prime factors: 81=3⁴. The square root of 8181 is the product of each factor raised to half its exponent: √81=3²=9.

3. Long Division Method

This traditional method involves dividing 81 into pairs (if needed) and finding the square root step by step. Since 81 is a perfect square, the method directly leads to 9.

4. Using a Calculator

The simplest and most accurate method for complex numbers. Inputting 81 into a calculator and pressing the square root function will yield 9.

5. Estimation Method

If you didn’t know 81 was a perfect square, you could estimate. Knowing 64 (8^2) is too low and 100 (10^2) is too high, you might guess the square root of 81 is between 8 and 10. With a bit of trial and error, or knowing it’s a perfect square, you’d arrive at 9.

Square Root of 81 by Long Division Method

Calculating Square Root by Long Division Method

Square Root of 81 (√81):

Recognize the operation: Understand that finding the square root of a number involves determining the number that, when multiplied by itself, equals the original number.

Compute: Calculate the square root of 81. Since 9×9 = 81, the square root of 81 is 9.

Interpretation: This means that the length of each side of a square with an area of 81 square units is 9 units.

Check: Verify the result by squaring the calculated square root. In this case, 9×9 = 81, confirming that the square root of 81 is indeed 9.

81 is Perfect Square root or Not

Yes, 81 is a perfect square

Because it is the square of 9 (9²=81). A perfect square is a number that can be expressed as the product of an integer with itself.


What does √ 81 mean?

The symbol √81 represents the square root of 81, which is the number that, when multiplied by itself, equals 81. For this specific case, the square root of 81 is 9.

Why is 81 squared rational?

81 squared is rational because the result, 6561, is an integer, and all integers are rational numbers, expressed as the quotient of the integer and one (6561/1).

What is 81 to cube root?

The cube root of 81 is approximately 4.3267. This value represents the number that, when multiplied by itself three times (cubed), equals 81.

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