Square Root of 150
Calculate √150 instantly — exact value, step-by-step explanation, and properties.
How to Calculate √150
The square root of 150 is approximately 12.2474487139. Since 150 is not a perfect square, its square root is an irrational number. We know that √144 = 12 and √169 = 13, so √150 falls between 12 and 13. The simplified radical form is 5√6. To estimate manually: start with 12.2, then refine using the Babylonian method — guess a value, divide 150 by it, average the result with your guess, and repeat.
The Babylonian Method (Step by Step)
- Start with an initial guess. Since √144 = 12 and √169 = 13, a good starting guess is 12.5.
- Divide 150 by your guess: 150 ÷ 12.5 = 12.0000
- Average the result with your guess: (12.5 + 12.0000) ÷ 2 = 12.250000
- Repeat with the new estimate until you reach the desired precision.
√150 ≈ 12.2474487139 Properties of √150
Perfect Square?
No
150 is not a perfect square. Its root is irrational.
Decimal Value
12.2474487139
Rounded to 10 decimal places.
Simplified Form
5√6
Simplified by extracting perfect square factors from the radicand.
Square Root Symbol
√150
Also written as 150^(1/2) or 150^0.5 in programming.
Related Square Root Values
Reference table of square roots near 150 for quick comparison.
Frequently Asked Questions
What is the square root of 150?
The square root of 150 is approximately 12.2474487139. This is an irrational number, which can be written in simplified radical form as 5√6.
Is 150 a perfect square?
No, 150 is not a perfect square. Its square root (12.2474487139...) is an irrational number that cannot be expressed as a simple fraction.
How do you calculate √150 by hand?
To calculate √150 by hand, use the Babylonian method: (1) Make an initial guess — since √144 = 12 and √169 = 13, start with 12.5. (2) Divide 150 by your guess. (3) Average the result with your guess. (4) Repeat steps 2-3 until you reach desired precision.
What is √150 in simplified radical form?
√150 can be simplified to 5√6. This is found by factoring out the largest perfect square from 150.