Square Root of 588

√588 at a glance

Exact value

14√3

Decimal (10 places)

24.2487113060

Rounded

24.2 · 24.25 · 24.249

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.248711

Prime factorization

2² × 3 × 7²

Cube root

8.377719

How to simplify √588

Look for the largest perfect square that divides 588. Here it is 196 (14²), because 588 = 196 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 588 = 2² × 3 × 7². Each pair of equal primes leaves the radical as one factor, so 2 × 7 comes out and 3 stays inside.

588 has 3 square factors (4, 49 and 196). Starting with a smaller one still works but takes more rounds: √588 = 2√147, and √147 can be simplified again. Using 196 straight away finishes in one step.

Check: (14√3)² = 14² × 3 = 196 × 3 = 588. As a decimal, 14√3 = 14 × 1.7320508076 ≈ 24.2487113060.

Where √588 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √588 lies between 24 and 25. 588 is 12 above 576 and 37 below 625, so the root is closer to 24.

• Straight line between 576 and 625: 24.2449 (0.02% low)
• Tangent from 24, i.e. 24 + 12 ÷ 48: 24.2500 (0.01% high)
• Tangent from 25, i.e. 25 − 37 ÷ 50: 24.2600 (0.05% high)

For √588 the tangent at 24 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 588 is just 12 above 576.

Finding √588 with the Babylonian method

Picture a rectangle with an area of 588 and one side x; the other side must be 588 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √588.

Start from the nearest whole number, 24 (24² = 576):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √588 = 24.2487113060 to every decimal shown.

√588 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √588 the pattern is [24; 4, 48] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √588 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 − 588y² = 1. Its smallest solution in positive whole numbers is x = 97, y = 4.

√588 in geometry and everyday measurements

• A square garage floor of 588 square feet measures about 24.25 ft (24 ft 3 in) per side, and its corner-to-corner diagonal is √1176 ≈ 34.3 ft.
• 588 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √588 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 10 × 22 box, because 2² + 10² + 22² = 588.
• Since √588 = 14√3, a length of √588 is exactly 14 copies of the length √3 laid end to end.

Square roots near √588 and related roots

• The cube root of 588 is about 8.377719.
• Because 588 = 4 × 147, the root is twice √147: 2 × 12.124356 ≈ 24.248711.

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