Square Root of 413

√413 at a glance

Exact value

√413

Decimal (10 places)

20.3224014329

Rounded

20.3 · 20.32 · 20.322

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.322401

Prime factorization

7 × 59

Cube root

7.447034

How to simplify √413

The prime factorization of 413 is 7 × 59. Every prime appears only once, so there is no pair to bring outside the radical — √413 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 413, 7 and 59 appear an odd number of times, so √413 is irrational and 20.3224014329 is a rounded value.

Where √413 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √413 lies between 20 and 21. 413 is 13 above 400 and 28 below 441, so the root is closer to 20.

• Straight line between 400 and 441: 20.3171 (0.03% low)
• Tangent from 20, i.e. 20 + 13 ÷ 40: 20.3250 (0.01% high)
• Tangent from 21, i.e. 21 − 28 ÷ 42: 20.3333 (0.05% high)

For √413 the tangent at 20 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 413 is just 13 above 400.

Finding √413 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 413: following the tangent line down to zero simplifies to averaging x with 413 ÷ x.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √413 = 20.3224014329 to every decimal shown.

√413 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √413 the pattern is [20; 3, 9, 1, 4, 1, 9, 3, 40] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √413 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 − 413y² = 1. Its smallest solution in positive whole numbers is x = 113,399, y = 5,580.

√413 in geometry and everyday measurements

• A square garage floor of 413 square feet measures about 20.32 ft (20 ft 4 in) per side, and its corner-to-corner diagonal is √826 ≈ 28.7 ft.
• 413 is not a sum of two whole-number squares — the prime factor 7 and 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √413 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 3 × 20 box, because 2² + 3² + 20² = 413.

Square roots near √413 and related roots

• The cube root of 413 is about 7.447034.
• Squaring undoes the root: (√413)² = 413, while 413² = 170,569 — the number whose square root is 413.

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