Square Root of 72

√72 at a glance

Exact value

6√2

Decimal (10 places)

8.4852813742

Rounded

8.5 · 8.49 · 8.485

Perfect square?

No — between 8² and 9²

Rational?

Irrational

Both square roots

±8.485281

Prime factorization

2³ × 3²

Cube root

4.160168

How to simplify √72

Look for the largest perfect square that divides 72. Here it is 36 (6²), because 72 = 36 × 2 and 2 has no square factor left:

The prime factorization tells the same story: 72 = 2³ × 3². Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 2 stays inside.

72 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √72 = 2√18, and √18 can be simplified again. Using 36 straight away finishes in one step.

Check: (6√2)² = 6² × 2 = 36 × 2 = 72. As a decimal, 6√2 = 6 × 1.4142135624 ≈ 8.4852813742.

Where √72 sits between perfect squares

64 = 8² and 81 = 9² are the nearest perfect squares, so √72 lies between 8 and 9. 72 is 8 above 64 and 9 below 81, so the root is closer to 8.

• Straight line between 64 and 81: 8.4706 (0.17% low)
• Tangent from 8, i.e. 8 + 8 ÷ 16: 8.5000 (0.17% high)
• Tangent from 9, i.e. 9 − 9 ÷ 18: 8.5000 (0.17% high)

For √72 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √72 with the Babylonian method

Picture a rectangle with an area of 72 and one side x; the other side must be 72 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √72.

Start from the nearest whole number, 8 (8² = 64):

The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √72 = 8.4852813742 to every decimal shown.

√72 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √72 the pattern is [8; 2, 16] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √72 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 72y² = 1. Its smallest solution in positive whole numbers is x = 17, y = 2.

√72 in geometry and everyday measurements

• A square room or garden bed covering 72 square feet measures about 8.49 ft (8 ft 6 in) along each wall.
• 72 = 6² + 6², so by the Pythagorean theorem √72 is the diagonal of a 6 × 6 rectangle — and the distance between the points (0, 0) and (6, 6) on a grid.
• Since √72 = 6√2, a length of √72 is exactly 6 copies of the length √2 laid end to end.

Square roots near √72 and related roots

• The cube root of 72 is about 4.160168.
• Four times the radicand doubles the root: √288 = 2 × √72 ≈ 16.970563.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.