Square Root of 401

√401 at a glance

Exact value

√401

Decimal (10 places)

20.0249843945

Rounded

20.0 · 20.02 · 20.025

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.024984

Prime factorization

401

Cube root

7.374198

How to simplify √401

401 is a prime number, so its only factors are 1 and 401. There is no perfect-square factor to pull out, which means √401 is already in its simplest radical form.

The square root of any prime is irrational. If √401 were a fraction a/b in lowest terms, then a² = 401b², so 401 would divide a — and then 401 would divide b too, contradicting “lowest terms.” That is why the decimal 20.0249843945 is only a rounded value.

Where √401 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √401 lies between 20 and 21. 401 is 1 above 400 and 40 below 441, so the root is closer to 20.

• Straight line between 400 and 441: 20.0244 (0% low)
• Tangent from 20, i.e. 20 + 1 ÷ 40: 20.0250 (0% high)
• Tangent from 21, i.e. 21 − 40 ÷ 42: 20.0476 (0.11% high)

For √401 the tangent at 20 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 401 is just 1 above 400.

Finding √401 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 401: following the tangent line down to zero simplifies to averaging x with 401 ÷ x.

Start from the nearest whole number, 20 (20² = 400):

Because the starting guess was already close, two steps are enough to match √401 = 20.0249843945 to every decimal shown.

√401 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √401 the pattern is [20; 40] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 401 is one more than a perfect square (20² + 1). A pattern that never ends is one more proof that √401 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 − 401y² = 1. Its smallest solution in positive whole numbers is x = 801, y = 40. Because the period is odd, the equation with −1 on the right also has a solution: 20² − 401 × 1² = −1.

√401 in geometry and everyday measurements

• A square garage floor of 401 square feet measures about 20.02 ft (20 ft) per side, and its corner-to-corner diagonal is √802 ≈ 28.3 ft.
• 401 = 1² + 20², so by the Pythagorean theorem √401 is the diagonal of a 1 × 20 rectangle — and the distance between the points (0, 0) and (1, 20) on a grid.

Square roots near √401 and related roots

• The cube root of 401 is about 7.374198.
• Squaring undoes the root: (√401)² = 401, while 401² = 160,801 — the number whose square root is 401.

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