Square Root of 461

√461 at a glance

Exact value

√461

Decimal (10 places)

21.4709105536

Rounded

21.5 · 21.47 · 21.471

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.470911

Prime factorization

461

Cube root

7.725032

How to simplify √461

461 is a prime number, so its only factors are 1 and 461. There is no perfect-square factor to pull out, which means √461 is already in its simplest radical form.

The square root of any prime is irrational. If √461 were a fraction a/b in lowest terms, then a² = 461b², so 461 would divide a — and then 461 would divide b too, contradicting “lowest terms.” That is why the decimal 21.4709105536 is only a rounded value.

Where √461 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √461 lies between 21 and 22. 461 is 20 above 441 and 23 below 484, so the root is closer to 21.

• Straight line between 441 and 484: 21.4651 (0.03% low)
• Tangent from 21, i.e. 21 + 20 ÷ 42: 21.4762 (0.02% high)
• Tangent from 22, i.e. 22 − 23 ÷ 44: 21.4773 (0.03% high)

For √461 the tangent at 21 wins, missing by only 0.0053. Tangent estimates shine when the number sits close to a perfect square — here 461 is just 20 above 441.

Finding √461 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 461: following the tangent line down to zero simplifies to averaging x with 461 ÷ 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 √461 = 21.4709105536 to every decimal shown.

√461 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √461 the pattern is [21; 2, 8, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, …] with the block of 15 terms after the semicolon repeating forever (only the first 12 of the 15 are shown). A pattern that never ends is one more proof that √461 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 − 461y² = 1. Its smallest solution in positive whole numbers is x = 1,182,351,890,184,201, y = 55,067,617,520,620 — 16 digits for x, even though 461 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: 24,314,110² − 461 × 1,132,421² = −1.

√461 in geometry and everyday measurements

• A square garage floor of 461 square feet measures about 21.47 ft (21 ft 6 in) per side, and its corner-to-corner diagonal is √922 ≈ 30.4 ft.
• 461 = 10² + 19², so by the Pythagorean theorem √461 is the diagonal of a 10 × 19 rectangle — and the distance between the points (0, 0) and (10, 19) on a grid.

Square roots near √461 and related roots

• The cube root of 461 is about 7.725032.
• Squaring undoes the root: (√461)² = 461, while 461² = 212,521 — the number whose square root is 461.

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