Square Root of 647

√647 at a glance

Exact value

√647

Decimal (10 places)

25.4361946840

Rounded

25.4 · 25.44 · 25.436

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.436195

Prime factorization

647

Cube root

8.649044

How to simplify √647

647 is a prime number, so its only factors are 1 and 647. There is no perfect-square factor to pull out, which means √647 is already in its simplest radical form.

The square root of any prime is irrational. If √647 were a fraction a/b in lowest terms, then a² = 647b², so 647 would divide a — and then 647 would divide b too, contradicting “lowest terms.” That is why the decimal 25.4361946840 is only a rounded value.

Where √647 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √647 lies between 25 and 26. 647 is 22 above 625 and 29 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.4314 (0.02% low)
• Tangent from 25, i.e. 25 + 22 ÷ 50: 25.4400 (0.01% high)
• Tangent from 26, i.e. 26 − 29 ÷ 52: 25.4423 (0.02% high)

For √647 the tangent at 25 wins, missing by only 0.0038. Tangent estimates shine when the number sits close to a perfect square — here 647 is just 22 above 625.

Finding √647 with the Babylonian method

If a guess is too big, 647 ÷ 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√647) in one step.

Start from the nearest whole number, 25 (25² = 625):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √647 = 25.4361946840 to every decimal shown.

√647 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √647 the pattern is [25; 2, 3, 2, 2, 1, 1, 4, 25, 4, 1, 1, 2, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √647 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 − 647y² = 1. Its smallest solution in positive whole numbers is x = 120,187,368, y = 4,725,053.

√647 in geometry and everyday measurements

• A square garage floor of 647 square feet measures about 25.44 ft (25 ft 5 in) per side, and its corner-to-corner diagonal is √1294 ≈ 36 ft.
• 647 is not a sum of two whole-number squares — 647 is itself a prime that is one less than a multiple of 4, which rules that out — so √647 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 √647 as its space diagonal.

Square roots near √647 and related roots

• The cube root of 647 is about 8.649044.
• Squaring undoes the root: (√647)² = 647, while 647² = 418,609 — the number whose square root is 647.

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