Square Root of 457

√457 at a glance

Exact value

√457

Decimal (10 places)

21.3775583264

Rounded

21.4 · 21.38 · 21.378

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.377558

Prime factorization

457

Cube root

7.702625

How to simplify √457

457 is a prime number, so its only factors are 1 and 457. There is no perfect-square factor to pull out, which means √457 is already in its simplest radical form.

The square root of any prime is irrational. If √457 were a fraction a/b in lowest terms, then a² = 457b², so 457 would divide a — and then 457 would divide b too, contradicting “lowest terms.” That is why the decimal 21.3775583264 is only a rounded value.

Where √457 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √457 lies between 21 and 22. 457 is 16 above 441 and 27 below 484, so the root is closer to 21.

• Straight line between 441 and 484: 21.3721 (0.03% low)
• Tangent from 21, i.e. 21 + 16 ÷ 42: 21.3810 (0.02% high)
• Tangent from 22, i.e. 22 − 27 ÷ 44: 21.3864 (0.04% high)

For √457 the tangent at 21 wins, missing by only 0.0034. Tangent estimates shine when the number sits close to a perfect square — here 457 is just 16 above 441.

Finding √457 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 457: following the tangent line down to zero simplifies to averaging x with 457 ÷ 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 √457 = 21.3775583264 to every decimal shown.

√457 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √457 the pattern is [21; 2, 1, 1, 1, 5, 2, 13, 1, 3, 1, 4, 1, …] with the block of 25 terms after the semicolon repeating forever (only the first 12 of the 25 are shown). A pattern that never ends is one more proof that √457 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 − 457y² = 1. Its smallest solution in positive whole numbers is x = 6,983,244,756,398,928,218,113, y = 326,662,411,570,389,853,632 — 22 digits for x, even though 457 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: 59,089,951,584² − 457 × 2,764,111,349² = −1.

√457 in geometry and everyday measurements

• A square garage floor of 457 square feet measures about 21.38 ft (21 ft 5 in) per side, and its corner-to-corner diagonal is √914 ≈ 30.2 ft.
• 457 = 4² + 21², so by the Pythagorean theorem √457 is the diagonal of a 4 × 21 rectangle — and the distance between the points (0, 0) and (4, 21) on a grid.

Square roots near √457 and related roots

• The cube root of 457 is about 7.702625.
• Squaring undoes the root: (√457)² = 457, while 457² = 208,849 — the number whose square root is 457.

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