Square Root of 10
Calculate √10 instantly — exact value, step-by-step explanation, and properties.
How to Calculate √10
The square root of 10 is approximately 3.1622776602. Since 10 is not a perfect square, its square root is an irrational number. We know that √9 = 3 and √16 = 4, so √10 falls between 3 and 4. To estimate manually: start with 3.1, then refine using the Babylonian method — guess a value, divide 10 by it, average the result with your guess, and repeat.
The Babylonian Method (Step by Step)
- Start with an initial guess. Since √9 = 3 and √16 = 4, a good starting guess is 3.5.
- Divide 10 by your guess: 10 ÷ 3.5 = 2.8571
- Average the result with your guess: (3.5 + 2.8571) ÷ 2 = 3.178571
- Repeat with the new estimate until you reach the desired precision.
√10 ≈ 3.1622776602 Properties of √10
Perfect Square?
No
10 is not a perfect square. Its root is irrational.
Decimal Value
3.1622776602
Rounded to 10 decimal places.
Square Root Symbol
√10
Also written as 10^(1/2) or 10^0.5 in programming.
Frequently Asked Questions
What is the square root of 10?
The square root of 10 is approximately 3.1622776602. This is an irrational number.
Is 10 a perfect square?
No, 10 is not a perfect square. Its square root (3.1622776602...) is an irrational number that cannot be expressed as a simple fraction.
How do you calculate √10 by hand?
To calculate √10 by hand, use the Babylonian method: (1) Make an initial guess — since √9 = 3 and √16 = 4, start with 3.5. (2) Divide 10 by your guess. (3) Average the result with your guess. (4) Repeat steps 2-3 until you reach desired precision.
What is √10 in simplified radical form?
√10 is already in its simplest radical form. The number 10 has no perfect square factors other than 1.