Square Root of 475

√475 at a glance

Exact value

5√19

Decimal (10 places)

21.7944947177

Rounded

21.8 · 21.79 · 21.794

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.794495

Prime factorization

5² × 19

Cube root

7.802454

How to simplify √475

Look for the largest perfect square that divides 475. Here it is 25 (5²), because 475 = 25 × 19 and 19 has no square factor left:

The prime factorization tells the same story: 475 = 5² × 19. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 19 stays inside.

Check: (5√19)² = 5² × 19 = 25 × 19 = 475. As a decimal, 5√19 = 5 × 4.3588989435 ≈ 21.7944947177.

Where √475 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √475 lies between 21 and 22. 475 is 34 above 441 and 9 below 484, so the root is closer to 22.

• Straight line between 441 and 484: 21.7907 (0.02% low)
• Tangent from 21, i.e. 21 + 34 ÷ 42: 21.8095 (0.07% high)
• Tangent from 22, i.e. 22 − 9 ÷ 44: 21.7955 (0% high)

For √475 the tangent at 22 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 475 is just 9 below 484.

Finding √475 with the Babylonian method

If a guess is too big, 475 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√475) in one step.

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 √475 = 21.7944947177 to every decimal shown.

√475 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √475 the pattern is [21; 1, 3, 1, 6, 2, 6, 1, 3, 1, 42] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √475 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 − 475y² = 1. Its smallest solution in positive whole numbers is x = 57,799, y = 2,652.

√475 in geometry and everyday measurements

• A square garage floor of 475 square feet measures about 21.79 ft (21 ft 10 in) per side, and its corner-to-corner diagonal is √950 ≈ 30.8 ft.
• 475 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √475 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 5 × 21 box, because 3² + 5² + 21² = 475.
• Since √475 = 5√19, a length of √475 is exactly 5 copies of the length √19 laid end to end.

Square roots near √475 and related roots

• The cube root of 475 is about 7.802454.
• Squaring undoes the root: (√475)² = 475, while 475² = 225,625 — the number whose square root is 475.

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