Square Root of 80

√80 at a glance

Exact value

4√5

Decimal (10 places)

8.9442719100

Rounded

8.9 · 8.94 · 8.944

Perfect square?

No — between 8² and 9²

Rational?

Irrational

Both square roots

±8.944272

Prime factorization

2⁴ × 5

Cube root

4.308869

How to simplify √80

Look for the largest perfect square that divides 80. Here it is 16 (4²), because 80 = 16 × 5 and 5 has no square factor left:

The prime factorization tells the same story: 80 = 2⁴ × 5. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 5 stays inside.

80 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √80 = 2√20, and √20 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√5)² = 4² × 5 = 16 × 5 = 80. As a decimal, 4√5 = 4 × 2.2360679775 ≈ 8.9442719100.

Where √80 sits between perfect squares

64 = 8² and 81 = 9² are the nearest perfect squares, so √80 lies between 8 and 9. 80 is 16 above 64 and 1 below 81, so the root is closer to 9.

• Straight line between 64 and 81: 8.9412 (0.03% low)
• Tangent from 8, i.e. 8 + 16 ÷ 16: 9.0000 (0.62% high)
• Tangent from 9, i.e. 9 − 1 ÷ 18: 8.9444 (0% high)

For √80 the tangent at 9 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 80 is just 1 below 81.

Finding √80 with the Babylonian method

Picture a rectangle with an area of 80 and one side x; the other side must be 80 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √80.

Start from the nearest whole number, 9 (9² = 81):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √80 = 8.9442719100 to every decimal shown.

√80 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √80 the pattern is [8; 1, 16] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √80 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 − 80y² = 1. Its smallest solution in positive whole numbers is x = 9, y = 1.

√80 in geometry and everyday measurements

• A square room or garden bed covering 80 square feet measures about 8.94 ft (8 ft 11 in) along each wall.
• 80 = 4² + 8², so by the Pythagorean theorem √80 is the diagonal of a 4 × 8 rectangle — and the distance between the points (0, 0) and (4, 8) on a grid.
• Since √80 = 4√5, a length of √80 is exactly 4 copies of the length √5 laid end to end.

Square roots near √80 and related roots

• The cube root of 80 is about 4.308869.
• Four times the radicand doubles the root: √320 = 2 × √80 ≈ 17.888544.

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