Square Root of 397

√397 at a glance

Exact value

√397

Decimal (10 places)

19.9248588452

Rounded

19.9 · 19.92 · 19.925

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.924859

Prime factorization

397

Cube root

7.349597

How to simplify √397

397 is a prime number, so its only factors are 1 and 397. There is no perfect-square factor to pull out, which means √397 is already in its simplest radical form.

The square root of any prime is irrational. If √397 were a fraction a/b in lowest terms, then a² = 397b², so 397 would divide a — and then 397 would divide b too, contradicting “lowest terms.” That is why the decimal 19.9248588452 is only a rounded value.

Where √397 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √397 lies between 19 and 20. 397 is 36 above 361 and 3 below 400, so the root is closer to 20.

• Straight line between 361 and 400: 19.9231 (0.01% low)
• Tangent from 19, i.e. 19 + 36 ÷ 38: 19.9474 (0.11% high)
• Tangent from 20, i.e. 20 − 3 ÷ 40: 19.9250 (0% high)

For √397 the tangent at 20 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 397 is just 3 below 400.

Finding √397 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 397: following the tangent line down to zero simplifies to averaging x with 397 ÷ x.

Start from the nearest whole number, 20 (20² = 400):

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √397 = 19.9248588452 to every decimal shown.

√397 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √397 the pattern is [19; 1, 12, 3, 4, 9, 1, 2, 1, 2, 1, 1, 2, …] with the block of 21 terms after the semicolon repeating forever (only the first 12 of the 21 are shown). A pattern that never ends is one more proof that √397 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 − 397y² = 1. Its smallest solution in positive whole numbers is x = 838,721,786,045,180,184,649, y = 42,094,239,791,738,433,660 — 21 digits for x, even though 397 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: 20,478,302,982² − 397 × 1,027,776,565² = −1.

√397 in geometry and everyday measurements

• A square patio or deck of 397 square feet is about 19.92 ft (19 ft 11 in) on each side, so edging all the way around takes 4 × √397 ≈ 79.7 ft.
• 397 = 6² + 19², so by the Pythagorean theorem √397 is the diagonal of a 6 × 19 rectangle — and the distance between the points (0, 0) and (6, 19) on a grid.

Square roots near √397 and related roots

• The cube root of 397 is about 7.349597.
• Squaring undoes the root: (√397)² = 397, while 397² = 157,609 — the number whose square root is 397.

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