Square Root of 389

√389 at a glance

Exact value

√389

Decimal (10 places)

19.7230829233

Rounded

19.7 · 19.72 · 19.723

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.723083

Prime factorization

389

Cube root

7.299894

How to simplify √389

389 is a prime number, so its only factors are 1 and 389. There is no perfect-square factor to pull out, which means √389 is already in its simplest radical form.

The square root of any prime is irrational. If √389 were a fraction a/b in lowest terms, then a² = 389b², so 389 would divide a — and then 389 would divide b too, contradicting “lowest terms.” That is why the decimal 19.7230829233 is only a rounded value.

Where √389 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √389 lies between 19 and 20. 389 is 28 above 361 and 11 below 400, so the root is closer to 20.

• Straight line between 361 and 400: 19.7179 (0.03% low)
• Tangent from 19, i.e. 19 + 28 ÷ 38: 19.7368 (0.07% high)
• Tangent from 20, i.e. 20 − 11 ÷ 40: 19.7250 (0.01% high)

For √389 the tangent at 20 wins, missing by only 0.0019. Tangent estimates shine when the number sits close to a perfect square — here 389 is just 11 below 400.

Finding √389 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 389: following the tangent line down to zero simplifies to averaging x with 389 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √389 = 19.7230829233 to every decimal shown.

√389 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √389 the pattern is [19; 1, 2, 1, 1, 1, 1, 2, 1, 38] with the block of 9 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √389 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 − 389y² = 1. Its smallest solution in positive whole numbers is x = 3,287,049, y = 166,660. Because the period is odd, the equation with −1 on the right also has a solution: 1,282² − 389 × 65² = −1.

√389 in geometry and everyday measurements

• A square patio or deck of 389 square feet is about 19.72 ft (19 ft 9 in) on each side, so edging all the way around takes 4 × √389 ≈ 78.9 ft.
• 389 = 10² + 17², so by the Pythagorean theorem √389 is the diagonal of a 10 × 17 rectangle — and the distance between the points (0, 0) and (10, 17) on a grid.

Square roots near √389 and related roots

• The cube root of 389 is about 7.299894.
• Squaring undoes the root: (√389)² = 389, while 389² = 151,321 — the number whose square root is 389.

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