Square Root of 388

√388 at a glance

Exact value

2√97

Decimal (10 places)

19.6977156036

Rounded

19.7 · 19.70 · 19.698

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.697716

Prime factorization

2² × 97

Cube root

7.293633

How to simplify √388

Look for the largest perfect square that divides 388. Here it is 4 (2²), because 388 = 4 × 97 and 97 has no square factor left:

The prime factorization tells the same story: 388 = 2² × 97. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 97 stays inside.

Check: (2√97)² = 2² × 97 = 4 × 97 = 388. As a decimal, 2√97 = 2 × 9.8488578018 ≈ 19.6977156036.

Where √388 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √388 lies between 19 and 20. 388 is 27 above 361 and 12 below 400, so the root is closer to 20.

• Straight line between 361 and 400: 19.6923 (0.03% low)
• Tangent from 19, i.e. 19 + 27 ÷ 38: 19.7105 (0.07% high)
• Tangent from 20, i.e. 20 − 12 ÷ 40: 19.7000 (0.01% high)

For √388 the tangent at 20 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 388 is just 12 below 400.

Finding √388 with the Babylonian method

Picture a rectangle with an area of 388 and one side x; the other side must be 388 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √388.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √388 = 19.6977156036 to every decimal shown.

√388 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √388 the pattern is [19; 1, 2, 3, 4, 12, 1, 8, 1, 12, 4, 3, 2, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √388 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 − 388y² = 1. Its smallest solution in positive whole numbers is x = 62,809,633, y = 3,188,676.

√388 in geometry and everyday measurements

• A square patio or deck of 388 square feet is about 19.7 ft (19 ft 8 in) on each side, so edging all the way around takes 4 × √388 ≈ 78.8 ft.
• 388 = 8² + 18², so by the Pythagorean theorem √388 is the diagonal of a 8 × 18 rectangle — and the distance between the points (0, 0) and (8, 18) on a grid.
• Since √388 = 2√97, a length of √388 is exactly 2 copies of the length √97 laid end to end.

Square roots near √388 and related roots

• The cube root of 388 is about 7.293633.
• Because 388 = 4 × 97, the root is twice √97: 2 × 9.848858 ≈ 19.697716.

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