Square Root of 1000
Calculate √1000 instantly — exact value, step-by-step explanation, and properties.
How to Calculate √1000
The square root of 1000 is approximately 31.6227766017. Since 1000 is not a perfect square, its square root is an irrational number. We know that √961 = 31 and √1024 = 32, so √1000 falls between 31 and 32. The simplified radical form is 10√10. To estimate manually: start with 31.6, then refine using the Babylonian method — guess a value, divide 1000 by it, average the result with your guess, and repeat.
The Babylonian Method (Step by Step)
- Start with an initial guess. Since √961 = 31 and √1024 = 32, a good starting guess is 31.5.
- Divide 1000 by your guess: 1000 ÷ 31.5 = 31.7460
- Average the result with your guess: (31.5 + 31.7460) ÷ 2 = 31.623016
- Repeat with the new estimate until you reach the desired precision.
√1000 ≈ 31.6227766017 Properties of √1000
Perfect Square?
No
1000 is not a perfect square. Its root is irrational.
Decimal Value
31.6227766017
Rounded to 10 decimal places.
Simplified Form
10√10
Simplified by extracting perfect square factors from the radicand.
Square Root Symbol
√1000
Also written as 1000^(1/2) or 1000^0.5 in programming.
Frequently Asked Questions
What is the square root of 1000?
The square root of 1000 is approximately 31.6227766017. This is an irrational number, which can be written in simplified radical form as 10√10.
Is 1000 a perfect square?
No, 1000 is not a perfect square. Its square root (31.6227766017...) is an irrational number that cannot be expressed as a simple fraction.
How do you calculate √1000 by hand?
To calculate √1000 by hand, use the Babylonian method: (1) Make an initial guess — since √961 = 31 and √1024 = 32, start with 31.5. (2) Divide 1000 by your guess. (3) Average the result with your guess. (4) Repeat steps 2-3 until you reach desired precision.
What is √1000 in simplified radical form?
√1000 can be simplified to 10√10. This is found by factoring out the largest perfect square from 1000.