Square Root of 857

√857 at a glance

Exact value

√857

Decimal (10 places)

29.2745623366

Rounded

29.3 · 29.27 · 29.275

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.274562

Prime factorization

857

Cube root

9.498615

How to simplify √857

857 is a prime number, so its only factors are 1 and 857. There is no perfect-square factor to pull out, which means √857 is already in its simplest radical form.

The square root of any prime is irrational. If √857 were a fraction a/b in lowest terms, then a² = 857b², so 857 would divide a — and then 857 would divide b too, contradicting “lowest terms.” That is why the decimal 29.2745623366 is only a rounded value.

Where √857 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √857 lies between 29 and 30. 857 is 16 above 841 and 43 below 900, so the root is closer to 29.

• Straight line between 841 and 900: 29.2712 (0.01% low)
• Tangent from 29, i.e. 29 + 16 ÷ 58: 29.2759 (0% high)
• Tangent from 30, i.e. 30 − 43 ÷ 60: 29.2833 (0.03% high)

For √857 the tangent at 29 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 857 is just 16 above 841.

Finding √857 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 857: following the tangent line down to zero simplifies to averaging x with 857 ÷ x.

Start from the nearest whole number, 29 (29² = 841):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √857 = 29.2745623366 to every decimal shown.

√857 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √857 the pattern is [29; 3, 1, 1, 1, 3, 1, 6, 1, 1, 6, 1, 3, …] with the block of 17 terms after the semicolon repeating forever (only the first 12 of the 17 are shown). A pattern that never ends is one more proof that √857 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 − 857y² = 1. Its smallest solution in positive whole numbers is x = 131,822,292,741,249, y = 4,502,963,741,200 — 15 digits for x, even though 857 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 8,118,568² − 857 × 277,325² = −1.

√857 in geometry and everyday measurements

• 857 square feet is 79.6 m². Laid out as a square — a small house footprint or a lot — it is about 29.27 ft (29 ft 3 in) on a side.
• 857 = 4² + 29², so by the Pythagorean theorem √857 is the diagonal of a 4 × 29 rectangle — and the distance between the points (0, 0) and (4, 29) on a grid.

Square roots near √857 and related roots

• The cube root of 857 is about 9.498615.
• Squaring undoes the root: (√857)² = 857, while 857² = 734,449 — the number whose square root is 857.

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