Square Root of 601

√601 at a glance

Exact value

√601

Decimal (10 places)

24.5153013443

Rounded

24.5 · 24.52 · 24.515

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.515301

Prime factorization

601

Cube root

8.439010

How to simplify √601

601 is a prime number, so its only factors are 1 and 601. There is no perfect-square factor to pull out, which means √601 is already in its simplest radical form.

The square root of any prime is irrational. If √601 were a fraction a/b in lowest terms, then a² = 601b², so 601 would divide a — and then 601 would divide b too, contradicting “lowest terms.” That is why the decimal 24.5153013443 is only a rounded value.

Where √601 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √601 lies between 24 and 25. 601 is 25 above 576 and 24 below 625, so the root is closer to 25.

• Straight line between 576 and 625: 24.5102 (0.02% low)
• Tangent from 24, i.e. 24 + 25 ÷ 48: 24.5208 (0.02% high)
• Tangent from 25, i.e. 25 − 24 ÷ 50: 24.5200 (0.02% high)

For √601 the tangent at 25 wins, missing by only 0.0047. Tangent estimates shine when the number sits close to a perfect square — here 601 is just 24 below 625.

Finding √601 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 601: following the tangent line down to zero simplifies to averaging x with 601 ÷ x.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √601 = 24.5153013443 to every decimal shown.

√601 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √601 the pattern is [24; 1, 1, 15, 1, 5, 5, 3, 1, 1, 2, 1, 2, …] with the block of 31 terms after the semicolon repeating forever (only the first 12 of the 31 are shown). A pattern that never ends is one more proof that √601 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 − 601y² = 1. Its smallest solution in positive whole numbers is x = 38,902,815,462,492,318,420,311,478,049, y = 1,586,878,942,101,888,360,258,625,080 — 29 digits for x, even though 601 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: 139,468,303,679,532² − 601 × 5,689,030,769,845² = −1.

√601 in geometry and everyday measurements

• A square garage floor of 601 square feet measures about 24.52 ft (24 ft 6 in) per side, and its corner-to-corner diagonal is √1202 ≈ 34.7 ft.
• 601 = 5² + 24², so by the Pythagorean theorem √601 is the diagonal of a 5 × 24 rectangle — and the distance between the points (0, 0) and (5, 24) on a grid.

Square roots near √601 and related roots

• The cube root of 601 is about 8.439010.
• Squaring undoes the root: (√601)² = 601, while 601² = 361,201 — the number whose square root is 601.

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