Square Root of 72
Calculate √72 instantly — exact value, step-by-step explanation, and properties.
How to Calculate √72
The square root of 72 is approximately 8.4852813742. Since 72 is not a perfect square, its square root is an irrational number. We know that √64 = 8 and √81 = 9, so √72 falls between 8 and 9. The simplified radical form is 6√2. To estimate manually: start with 8.4, then refine using the Babylonian method — guess a value, divide 72 by it, average the result with your guess, and repeat.
The Babylonian Method (Step by Step)
- Start with an initial guess. Since √64 = 8 and √81 = 9, a good starting guess is 8.5.
- Divide 72 by your guess: 72 ÷ 8.5 = 8.4706
- Average the result with your guess: (8.5 + 8.4706) ÷ 2 = 8.485294
- Repeat with the new estimate until you reach the desired precision.
√72 ≈ 8.4852813742 Properties of √72
Perfect Square?
No
72 is not a perfect square. Its root is irrational.
Decimal Value
8.4852813742
Rounded to 10 decimal places.
Simplified Form
6√2
Simplified by extracting perfect square factors from the radicand.
Square Root Symbol
√72
Also written as 72^(1/2) or 72^0.5 in programming.
Related Square Root Values
Reference table of square roots near 72 for quick comparison.
Frequently Asked Questions
What is the square root of 72?
The square root of 72 is approximately 8.4852813742. This is an irrational number, which can be written in simplified radical form as 6√2.
Is 72 a perfect square?
No, 72 is not a perfect square. Its square root (8.4852813742...) is an irrational number that cannot be expressed as a simple fraction.
How do you calculate √72 by hand?
To calculate √72 by hand, use the Babylonian method: (1) Make an initial guess — since √64 = 8 and √81 = 9, start with 8.5. (2) Divide 72 by your guess. (3) Average the result with your guess. (4) Repeat steps 2-3 until you reach desired precision.
What is √72 in simplified radical form?
√72 can be simplified to 6√2. This is found by factoring out the largest perfect square from 72.