Square Root of 500
Calculate √500 instantly — exact value, step-by-step explanation, and properties.
How to Calculate √500
The square root of 500 is approximately 22.3606797750. Since 500 is not a perfect square, its square root is an irrational number. We know that √484 = 22 and √529 = 23, so √500 falls between 22 and 23. The simplified radical form is 10√5. To estimate manually: start with 22.3, then refine using the Babylonian method — guess a value, divide 500 by it, average the result with your guess, and repeat.
The Babylonian Method (Step by Step)
- Start with an initial guess. Since √484 = 22 and √529 = 23, a good starting guess is 22.5.
- Divide 500 by your guess: 500 ÷ 22.5 = 22.2222
- Average the result with your guess: (22.5 + 22.2222) ÷ 2 = 22.361111
- Repeat with the new estimate until you reach the desired precision.
√500 ≈ 22.3606797750 Properties of √500
Perfect Square?
No
500 is not a perfect square. Its root is irrational.
Decimal Value
22.3606797750
Rounded to 10 decimal places.
Simplified Form
10√5
Simplified by extracting perfect square factors from the radicand.
Square Root Symbol
√500
Also written as 500^(1/2) or 500^0.5 in programming.
Frequently Asked Questions
What is the square root of 500?
The square root of 500 is approximately 22.3606797750. This is an irrational number, which can be written in simplified radical form as 10√5.
Is 500 a perfect square?
No, 500 is not a perfect square. Its square root (22.3606797750...) is an irrational number that cannot be expressed as a simple fraction.
How do you calculate √500 by hand?
To calculate √500 by hand, use the Babylonian method: (1) Make an initial guess — since √484 = 22 and √529 = 23, start with 22.5. (2) Divide 500 by your guess. (3) Average the result with your guess. (4) Repeat steps 2-3 until you reach desired precision.
What is √500 in simplified radical form?
√500 can be simplified to 10√5. This is found by factoring out the largest perfect square from 500.