Square Root of 907

√907 at a glance

Exact value

√907

Decimal (10 places)

30.1164406928

Rounded

30.1 · 30.12 · 30.116

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.116441

Prime factorization

907

Cube root

9.679860

How to simplify √907

907 is a prime number, so its only factors are 1 and 907. There is no perfect-square factor to pull out, which means √907 is already in its simplest radical form.

The square root of any prime is irrational. If √907 were a fraction a/b in lowest terms, then a² = 907b², so 907 would divide a — and then 907 would divide b too, contradicting “lowest terms.” That is why the decimal 30.1164406928 is only a rounded value.

Where √907 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √907 lies between 30 and 31. 907 is 7 above 900 and 54 below 961, so the root is closer to 30.

• Straight line between 900 and 961: 30.1148 (0.01% low)
• Tangent from 30, i.e. 30 + 7 ÷ 60: 30.1167 (0% high)
• Tangent from 31, i.e. 31 − 54 ÷ 62: 30.1290 (0.04% high)

For √907 the tangent at 30 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 907 is just 7 above 900.

Finding √907 with the Babylonian method

If a guess is too big, 907 ÷ 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√907) in one step.

Start from the nearest whole number, 30 (30² = 900):

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √907 = 30.1164406928 to every decimal shown.

√907 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √907 the pattern is [30; 8, 1, 1, 2, 2, 1, 19, 2, 1, 2, 5, 9, …] with the block of 42 terms after the semicolon repeating forever (only the first 12 of the 42 are shown). A pattern that never ends is one more proof that √907 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 − 907y² = 1. Its smallest solution in positive whole numbers is x = 123,823,410,343,073,497,682, y = 4,111,488,857,741,309,517 — 21 digits for x, even though 907 is small, which is what makes Pell’s equation famous.

√907 in geometry and everyday measurements

• 907 square feet is 84.3 m². Laid out as a square — a small house footprint or a lot — it is about 30.12 ft (30 ft 1 in) on a side.
• 907 is not a sum of two whole-number squares — 907 is itself a prime that is one less than a multiple of 4, which rules that out — so √907 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 13 × 27 box, because 3² + 13² + 27² = 907.

Square roots near √907 and related roots

• The cube root of 907 is about 9.679860.
• Squaring undoes the root: (√907)² = 907, while 907² = 822,649 — the number whose square root is 907.

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