Square Root of 476

√476 at a glance

Exact value

2√119

Decimal (10 places)

21.8174242293

Rounded

21.8 · 21.82 · 21.817

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.817424

Prime factorization

2² × 7 × 17

Cube root

7.807925

How to simplify √476

Look for the largest perfect square that divides 476. Here it is 4 (2²), because 476 = 4 × 119 and 119 has no square factor left:

The prime factorization tells the same story: 476 = 2² × 7 × 17. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 7 × 17 stays inside.

Check: (2√119)² = 2² × 119 = 4 × 119 = 476. As a decimal, 2√119 = 2 × 10.9087121146 ≈ 21.8174242293.

Where √476 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √476 lies between 21 and 22. 476 is 35 above 441 and 8 below 484, so the root is closer to 22.

• Straight line between 441 and 484: 21.8140 (0.02% low)
• Tangent from 21, i.e. 21 + 35 ÷ 42: 21.8333 (0.07% high)
• Tangent from 22, i.e. 22 − 8 ÷ 44: 21.8182 (0% high)

For √476 the tangent at 22 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 476 is just 8 below 484.

Finding √476 with the Babylonian method

Picture a rectangle with an area of 476 and one side x; the other side must be 476 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √476.

Start from the nearest whole number, 22 (22² = 484):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √476 = 21.8174242293 to every decimal shown.

√476 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √476 the pattern is [21; 1, 4, 2, 10, 2, 4, 1, 42] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √476 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 − 476y² = 1. Its smallest solution in positive whole numbers is x = 28,799, y = 1,320.

√476 in geometry and everyday measurements

• A square garage floor of 476 square feet measures about 21.82 ft (21 ft 10 in) per side, and its corner-to-corner diagonal is √952 ≈ 30.9 ft.
• 476 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √476 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √476 as its space diagonal.
• Since √476 = 2√119, a length of √476 is exactly 2 copies of the length √119 laid end to end.

Square roots near √476 and related roots

• The cube root of 476 is about 7.807925.
• Because 476 = 4 × 119, the root is twice √119: 2 × 10.908712 ≈ 21.817424.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.