Square Root of 71

√71 at a glance

Exact value

√71

Decimal (10 places)

8.4261497732

Rounded

8.4 · 8.43 · 8.426

Perfect square?

No — between 8² and 9²

Rational?

Irrational

Both square roots

±8.426150

Prime factorization

71

Cube root

4.140818

How to simplify √71

71 is a prime number, so its only factors are 1 and 71. There is no perfect-square factor to pull out, which means √71 is already in its simplest radical form.

The square root of any prime is irrational. If √71 were a fraction a/b in lowest terms, then a² = 71b², so 71 would divide a — and then 71 would divide b too, contradicting “lowest terms.” That is why the decimal 8.4261497732 is only a rounded value.

Where √71 sits between perfect squares

64 = 8² and 81 = 9² are the nearest perfect squares, so √71 lies between 8 and 9. 71 is 7 above 64 and 10 below 81, so the root is closer to 8.

• Straight line between 64 and 81: 8.4118 (0.17% low)
• Tangent from 8, i.e. 8 + 7 ÷ 16: 8.4375 (0.13% high)
• Tangent from 9, i.e. 9 − 10 ÷ 18: 8.4444 (0.22% high)

For √71 the tangent at 8 wins, missing by only 0.0114. Tangent estimates shine when the number sits close to a perfect square — here 71 is just 7 above 64.

Finding √71 with the Babylonian method

If a guess is too big, 71 ÷ 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√71) in one step.

Start from the nearest whole number, 8 (8² = 64):

The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √71 = 8.4261497732 to every decimal shown.

√71 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √71 the pattern is [8; 2, 2, 1, 7, 1, 2, 2, 16] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √71 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 − 71y² = 1. Its smallest solution in positive whole numbers is x = 3,480, y = 413.

√71 in geometry and everyday measurements

• A square room or garden bed covering 71 square feet measures about 8.43 ft (8 ft 5 in) along each wall.
• 71 is not a sum of two whole-number squares — 71 is itself a prime that is one less than a multiple of 4, which rules that out — so √71 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 √71 as its space diagonal.

Square roots near √71 and related roots

• The cube root of 71 is about 4.140818.
• Four times the radicand doubles the root: √284 = 2 × √71 ≈ 16.8523.

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