Square Root of 670

√670 at a glance

Exact value

√670

Decimal (10 places)

25.8843582111

Rounded

25.9 · 25.88 · 25.884

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.884358

Prime factorization

2 × 5 × 67

Cube root

8.750340

How to simplify √670

The prime factorization of 670 is 2 × 5 × 67. Every prime appears only once, so there is no pair to bring outside the radical — √670 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 670, 2, 5 and 67 appear an odd number of times, so √670 is irrational and 25.8843582111 is a rounded value.

Where √670 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √670 lies between 25 and 26. 670 is 45 above 625 and 6 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.8824 (0.01% low)
• Tangent from 25, i.e. 25 + 45 ÷ 50: 25.9000 (0.06% high)
• Tangent from 26, i.e. 26 − 6 ÷ 52: 25.8846 (0% high)

For √670 the tangent at 26 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 670 is just 6 below 676.

Finding √670 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, 26 (26² = 676):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √670 = 25.8843582111 to every decimal shown.

√670 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √670 the pattern is [25; 1, 7, 1, 1, 1, 5, 10, 5, 1, 1, 1, 7, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √670 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 − 670y² = 1. Its smallest solution in positive whole numbers is x = 5,791,211, y = 223,734.

√670 in geometry and everyday measurements

• 670 square feet is 62.2 m². Laid out as a square — a small house footprint or a lot — it is about 25.88 ft (25 ft 11 in) on a side.
• 670 is not a sum of two whole-number squares — the prime factor 67 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √670 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 15 × 21 box, because 2² + 15² + 21² = 670.

Square roots near √670 and related roots

• The cube root of 670 is about 8.750340.
• Squaring undoes the root: (√670)² = 670, while 670² = 448,900 — the number whose square root is 670.

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