Square Root of 725

√725 at a glance

Exact value

5√29

Decimal (10 places)

26.9258240357

Rounded

26.9 · 26.93 · 26.926

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.925824

Prime factorization

5² × 29

Cube root

8.983509

How to simplify √725

Look for the largest perfect square that divides 725. Here it is 25 (5²), because 725 = 25 × 29 and 29 has no square factor left:

The prime factorization tells the same story: 725 = 5² × 29. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 29 stays inside.

Check: (5√29)² = 5² × 29 = 25 × 29 = 725. As a decimal, 5√29 = 5 × 5.3851648071 ≈ 26.9258240357.

Where √725 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √725 lies between 26 and 27. 725 is 49 above 676 and 4 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.9245 (0% low)
• Tangent from 26, i.e. 26 + 49 ÷ 52: 26.9423 (0.06% high)
• Tangent from 27, i.e. 27 − 4 ÷ 54: 26.9259 (0% high)

For √725 the tangent at 27 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 725 is just 4 below 729.

Finding √725 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 725: following the tangent line down to zero simplifies to averaging x with 725 ÷ x.

Start from the nearest whole number, 27 (27² = 729):

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √725 = 26.9258240357 to every decimal shown.

√725 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √725 the pattern is [26; 1, 12, 2, 12, 1, 52] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √725 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 − 725y² = 1. Its smallest solution in positive whole numbers is x = 9,801, y = 364.

√725 in geometry and everyday measurements

• 725 square feet is 67.4 m². Laid out as a square — a small house footprint or a lot — it is about 26.93 ft (26 ft 11 in) on a side.
• 725 = 7² + 26² = 10² + 25² = 14² + 23², so by the Pythagorean theorem √725 is the diagonal of rectangles measuring 7 × 26, 10 × 25 and 14 × 23 — and the distance between the points (0, 0) and (7, 26) on a grid.
• Since √725 = 5√29, a length of √725 is exactly 5 copies of the length √29 laid end to end.

Square roots near √725 and related roots

• The cube root of 725 is about 8.983509.
• Squaring undoes the root: (√725)² = 725, while 725² = 525,625 — the number whose square root is 725.

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