Square Root of 672

√672 at a glance

Exact value

4√42

Decimal (10 places)

25.9229627936

Rounded

25.9 · 25.92 · 25.923

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.922963

Prime factorization

2⁵ × 3 × 7

Cube root

8.759038

How to simplify √672

Look for the largest perfect square that divides 672. Here it is 16 (4²), because 672 = 16 × 42 and 42 has no square factor left:

The prime factorization tells the same story: 672 = 2⁵ × 3 × 7. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 2 × 3 × 7 stays inside.

672 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √672 = 2√168, and √168 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√42)² = 4² × 42 = 16 × 42 = 672. As a decimal, 4√42 = 4 × 6.4807406984 ≈ 25.9229627936.

Where √672 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √672 lies between 25 and 26. 672 is 47 above 625 and 4 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.9216 (0.01% low)
• Tangent from 25, i.e. 25 + 47 ÷ 50: 25.9400 (0.07% high)
• Tangent from 26, i.e. 26 − 4 ÷ 52: 25.9231 (0% high)

For √672 the tangent at 26 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 672 is just 4 below 676.

Finding √672 with the Babylonian method

Picture a rectangle with an area of 672 and one side x; the other side must be 672 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √672.

Start from the nearest whole number, 26 (26² = 676):

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √672 = 25.9229627936 to every decimal shown.

√672 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √672 the pattern is [25; 1, 11, 1, 50] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √672 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 − 672y² = 1. Its smallest solution in positive whole numbers is x = 337, y = 13.

√672 in geometry and everyday measurements

• 672 square feet is 62.4 m². Laid out as a square — a small house footprint or a lot — it is about 25.92 ft (25 ft 11 in) on a side.
• 672 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √672 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 16 × 20 box, because 4² + 16² + 20² = 672.
• Since √672 = 4√42, a length of √672 is exactly 4 copies of the length √42 laid end to end.

Square roots near √672 and related roots

• The cube root of 672 is about 8.759038.
• Because 672 = 4 × 168, the root is twice √168: 2 × 12.961481 ≈ 25.922963.

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