Square Root of 421

√421 at a glance

Exact value

√421

Decimal (10 places)

20.5182845287

Rounded

20.5 · 20.52 · 20.518

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.518285

Prime factorization

421

Cube root

7.494811

How to simplify √421

421 is a prime number, so its only factors are 1 and 421. There is no perfect-square factor to pull out, which means √421 is already in its simplest radical form.

The square root of any prime is irrational. If √421 were a fraction a/b in lowest terms, then a² = 421b², so 421 would divide a — and then 421 would divide b too, contradicting “lowest terms.” That is why the decimal 20.5182845287 is only a rounded value.

Where √421 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √421 lies between 20 and 21. 421 is 21 above 400 and 20 below 441, so the root is closer to 21.

• Straight line between 400 and 441: 20.5122 (0.03% low)
• Tangent from 20, i.e. 20 + 21 ÷ 40: 20.5250 (0.03% high)
• Tangent from 21, i.e. 21 − 20 ÷ 42: 20.5238 (0.03% high)

For √421 the tangent at 21 wins, missing by only 0.0055. Tangent estimates shine when the number sits close to a perfect square — here 421 is just 20 below 441.

Finding √421 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 421: following the tangent line down to zero simplifies to averaging x with 421 ÷ x.

Start from the nearest whole number, 21 (21² = 441):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √421 = 20.5182845287 to every decimal shown.

√421 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √421 the pattern is [20; 1, 1, 13, 5, 1, 3, 1, 2, 1, 1, 1, 2, …] with the block of 37 terms after the semicolon repeating forever (only the first 12 of the 37 are shown). A pattern that never ends is one more proof that √421 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 − 421y² = 1. Its smallest solution in positive whole numbers is x = 3,879,474,045,914,926,879,468,217,167,061,449, y = 189,073,995,951,839,020,880,499,780,706,260 — 34 digits for x, even though 421 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: 44,042,445,696,821,418² − 421 × 2,146,497,463,530,785² = −1.

√421 in geometry and everyday measurements

• A square garage floor of 421 square feet measures about 20.52 ft (20 ft 6 in) per side, and its corner-to-corner diagonal is √842 ≈ 29 ft.
• 421 = 14² + 15², so by the Pythagorean theorem √421 is the diagonal of a 14 × 15 rectangle — and the distance between the points (0, 0) and (14, 15) on a grid.

Square roots near √421 and related roots

• The cube root of 421 is about 7.494811.
• Squaring undoes the root: (√421)² = 421, while 421² = 177,241 — the number whose square root is 421.

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