Square Root of 617

√617 at a glance

Exact value

√617

Decimal (10 places)

24.8394846967

Rounded

24.8 · 24.84 · 24.839

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.839485

Prime factorization

617

Cube root

8.513243

How to simplify √617

617 is a prime number, so its only factors are 1 and 617. There is no perfect-square factor to pull out, which means √617 is already in its simplest radical form.

The square root of any prime is irrational. If √617 were a fraction a/b in lowest terms, then a² = 617b², so 617 would divide a — and then 617 would divide b too, contradicting “lowest terms.” That is why the decimal 24.8394846967 is only a rounded value.

Where √617 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √617 lies between 24 and 25. 617 is 41 above 576 and 8 below 625, so the root is closer to 25.

• Straight line between 576 and 625: 24.8367 (0.01% low)
• Tangent from 24, i.e. 24 + 41 ÷ 48: 24.8542 (0.06% high)
• Tangent from 25, i.e. 25 − 8 ÷ 50: 24.8400 (0% high)

For √617 the tangent at 25 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 617 is just 8 below 625.

Finding √617 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 617: following the tangent line down to zero simplifies to averaging x with 617 ÷ x.

Start from the nearest whole number, 25 (25² = 625):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √617 = 24.8394846967 to every decimal shown.

√617 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √617 the pattern is [24; 1, 5, 4, 2, 1, 6, 2, 2, 6, 1, 2, 4, …] with the block of 15 terms after the semicolon repeating forever (only the first 12 of the 15 are shown). A pattern that never ends is one more proof that √617 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 − 617y² = 1. Its smallest solution in positive whole numbers is x = 3,363,593,612,801,313, y = 135,413,180,018,248 — 16 digits for x, even though 617 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 41,009,716² − 617 × 1,650,989² = −1.

√617 in geometry and everyday measurements

• A square garage floor of 617 square feet measures about 24.84 ft (24 ft 10 in) per side, and its corner-to-corner diagonal is √1234 ≈ 35.1 ft.
• 617 = 16² + 19², so by the Pythagorean theorem √617 is the diagonal of a 16 × 19 rectangle — and the distance between the points (0, 0) and (16, 19) on a grid.

Square roots near √617 and related roots

• The cube root of 617 is about 8.513243.
• Squaring undoes the root: (√617)² = 617, while 617² = 380,689 — the number whose square root is 617.

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