Square Root of 613

√613 at a glance

Exact value

√613

Decimal (10 places)

24.7588368063

Rounded

24.8 · 24.76 · 24.759

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.758837

Prime factorization

613

Cube root

8.494807

How to simplify √613

613 is a prime number, so its only factors are 1 and 613. There is no perfect-square factor to pull out, which means √613 is already in its simplest radical form.

The square root of any prime is irrational. If √613 were a fraction a/b in lowest terms, then a² = 613b², so 613 would divide a — and then 613 would divide b too, contradicting “lowest terms.” That is why the decimal 24.7588368063 is only a rounded value.

Where √613 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √613 lies between 24 and 25. 613 is 37 above 576 and 12 below 625, so the root is closer to 25.

• Straight line between 576 and 625: 24.7551 (0.02% low)
• Tangent from 24, i.e. 24 + 37 ÷ 48: 24.7708 (0.05% high)
• Tangent from 25, i.e. 25 − 12 ÷ 50: 24.7600 (0% high)

For √613 the tangent at 25 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 613 is just 12 below 625.

Finding √613 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 613: following the tangent line down to zero simplifies to averaging x with 613 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √613 = 24.7588368063 to every decimal shown.

√613 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √613 the pattern is [24; 1, 3, 6, 1, 4, 1, 1, 1, 3, 2, 11, 1, …] with the block of 33 terms after the semicolon repeating forever (only the first 12 of the 33 are shown). A pattern that never ends is one more proof that √613 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 − 613y² = 1. Its smallest solution in positive whole numbers is x = 464,018,873,584,078,278,910,994,299,849, y = 18,741,545,784,831,997,880,308,784,340 — 30 digits for x, even though 613 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: 481,673,579,088,618² − 613 × 19,454,612,624,065² = −1.

√613 in geometry and everyday measurements

• A square garage floor of 613 square feet measures about 24.76 ft (24 ft 9 in) per side, and its corner-to-corner diagonal is √1226 ≈ 35 ft.
• 613 = 17² + 18², so by the Pythagorean theorem √613 is the diagonal of a 17 × 18 rectangle — and the distance between the points (0, 0) and (17, 18) on a grid.

Square roots near √613 and related roots

• The cube root of 613 is about 8.494807.
• Squaring undoes the root: (√613)² = 613, while 613² = 375,769 — the number whose square root is 613.

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