Square Root of 125
Calculate √125 instantly — exact value, step-by-step explanation, and properties.
How to Calculate √125
The square root of 125 is approximately 11.1803398875. Since 125 is not a perfect square, its square root is an irrational number. We know that √121 = 11 and √144 = 12, so √125 falls between 11 and 12. The simplified radical form is 5√5. To estimate manually: start with 11.1, then refine using the Babylonian method — guess a value, divide 125 by it, average the result with your guess, and repeat.
The Babylonian Method (Step by Step)
- Start with an initial guess. Since √121 = 11 and √144 = 12, a good starting guess is 11.5.
- Divide 125 by your guess: 125 ÷ 11.5 = 10.8696
- Average the result with your guess: (11.5 + 10.8696) ÷ 2 = 11.184783
- Repeat with the new estimate until you reach the desired precision.
√125 ≈ 11.1803398875 Properties of √125
Perfect Square?
No
125 is not a perfect square. Its root is irrational.
Decimal Value
11.1803398875
Rounded to 10 decimal places.
Simplified Form
5√5
Simplified by extracting perfect square factors from the radicand.
Square Root Symbol
√125
Also written as 125^(1/2) or 125^0.5 in programming.
Related Square Root Values
Reference table of square roots near 125 for quick comparison.
Frequently Asked Questions
What is the square root of 125?
The square root of 125 is approximately 11.1803398875. This is an irrational number, which can be written in simplified radical form as 5√5.
Is 125 a perfect square?
No, 125 is not a perfect square. Its square root (11.1803398875...) is an irrational number that cannot be expressed as a simple fraction.
How do you calculate √125 by hand?
To calculate √125 by hand, use the Babylonian method: (1) Make an initial guess — since √121 = 11 and √144 = 12, start with 11.5. (2) Divide 125 by your guess. (3) Average the result with your guess. (4) Repeat steps 2-3 until you reach desired precision.
What is √125 in simplified radical form?
√125 can be simplified to 5√5. This is found by factoring out the largest perfect square from 125.