Square Root of 270

√270 at a glance

Exact value

3√30

Decimal (10 places)

16.4316767252

Rounded

16.4 · 16.43 · 16.432

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.431677

Prime factorization

2 × 3³ × 5

Cube root

6.463304

How to simplify √270

Look for the largest perfect square that divides 270. Here it is 9 (3²), because 270 = 9 × 30 and 30 has no square factor left:

The prime factorization tells the same story: 270 = 2 × 3³ × 5. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 3 × 5 stays inside.

Check: (3√30)² = 3² × 30 = 9 × 30 = 270. As a decimal, 3√30 = 3 × 5.4772255751 ≈ 16.4316767252.

Where √270 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √270 lies between 16 and 17. 270 is 14 above 256 and 19 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.4242 (0.05% low)
• Tangent from 16, i.e. 16 + 14 ÷ 32: 16.4375 (0.04% high)
• Tangent from 17, i.e. 17 − 19 ÷ 34: 16.4412 (0.06% high)

For √270 the tangent at 16 wins, missing by only 0.0058. Tangent estimates shine when the number sits close to a perfect square — here 270 is just 14 above 256.

Finding √270 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 16 (16² = 256):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √270 = 16.4316767252 to every decimal shown.

√270 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √270 the pattern is [16; 2, 3, 6, 3, 2, 32] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √270 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 − 270y² = 1. Its smallest solution in positive whole numbers is x = 5,291, y = 322.

√270 in geometry and everyday measurements

• A square patio or deck of 270 square feet is about 16.43 ft (16 ft 5 in) on each side, so edging all the way around takes 4 × √270 ≈ 65.7 ft.
• 270 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √270 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 10 × 13 box, because 1² + 10² + 13² = 270.
• Since √270 = 3√30, a length of √270 is exactly 3 copies of the length √30 laid end to end.

Square roots near √270 and related roots

• The cube root of 270 is about 6.463304.
• Squaring undoes the root: (√270)² = 270, while 270² = 72,900 — the number whose square root is 270.

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