Square Root of 47

√47 at a glance

Exact value

√47

Decimal (10 places)

6.8556546004

Rounded

6.9 · 6.86 · 6.856

Perfect square?

No — between 6² and 7²

Rational?

Irrational

Both square roots

±6.855655

Prime factorization

47

Cube root

3.608826

How to simplify √47

47 is a prime number, so its only factors are 1 and 47. There is no perfect-square factor to pull out, which means √47 is already in its simplest radical form.

The square root of any prime is irrational. If √47 were a fraction a/b in lowest terms, then a² = 47b², so 47 would divide a — and then 47 would divide b too, contradicting “lowest terms.” That is why the decimal 6.8556546004 is only a rounded value.

Where √47 sits between perfect squares

36 = 6² and 49 = 7² are the nearest perfect squares, so √47 lies between 6 and 7. 47 is 11 above 36 and 2 below 49, so the root is closer to 7.

• Straight line between 36 and 49: 6.8462 (0.14% low)
• Tangent from 6, i.e. 6 + 11 ÷ 12: 6.9167 (0.89% high)
• Tangent from 7, i.e. 7 − 2 ÷ 14: 6.8571 (0.02% high)

For √47 the tangent at 7 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 47 is just 2 below 49.

Finding √47 with the Babylonian method

If a guess is too big, 47 ÷ 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√47) in one step.

Start from the nearest whole number, 7 (7² = 49):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √47 = 6.8556546004 to every decimal shown.

√47 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √47 the pattern is [6; 1, 5, 1, 12] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √47 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 − 47y² = 1. Its smallest solution in positive whole numbers is x = 48, y = 7.

√47 in geometry and everyday measurements

• A square room or garden bed covering 47 square feet measures about 6.86 ft (6 ft 10 in) along each wall.
• 47 is not a sum of two whole-number squares — 47 is itself a prime that is one less than a multiple of 4, which rules that out — so √47 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 √47 as its space diagonal.

Square roots near √47 and related roots

• The cube root of 47 is about 3.608826.
• Four times the radicand doubles the root: √188 = 2 × √47 ≈ 13.711309.

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