Square Root of 584

√584 at a glance

Exact value

2√146

Decimal (10 places)

24.1660919472

Rounded

24.2 · 24.17 · 24.166

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.166092

Prime factorization

2³ × 73

Cube root

8.358678

How to simplify √584

Look for the largest perfect square that divides 584. Here it is 4 (2²), because 584 = 4 × 146 and 146 has no square factor left:

The prime factorization tells the same story: 584 = 2³ × 73. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 73 stays inside.

Check: (2√146)² = 2² × 146 = 4 × 146 = 584. As a decimal, 2√146 = 2 × 12.0830459736 ≈ 24.1660919472.

Where √584 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √584 lies between 24 and 25. 584 is 8 above 576 and 41 below 625, so the root is closer to 24.

• Straight line between 576 and 625: 24.1633 (0.01% low)
• Tangent from 24, i.e. 24 + 8 ÷ 48: 24.1667 (0% high)
• Tangent from 25, i.e. 25 − 41 ÷ 50: 24.1800 (0.06% high)

For √584 the tangent at 24 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 584 is just 8 above 576.

Finding √584 with the Babylonian method

Picture a rectangle with an area of 584 and one side x; the other side must be 584 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √584.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √584 = 24.1660919472 to every decimal shown.

√584 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √584 the pattern is [24; 6, 48] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √584 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 − 584y² = 1. Its smallest solution in positive whole numbers is x = 145, y = 6.

√584 in geometry and everyday measurements

• A square garage floor of 584 square feet measures about 24.17 ft (24 ft 2 in) per side, and its corner-to-corner diagonal is √1168 ≈ 34.2 ft.
• 584 = 10² + 22², so by the Pythagorean theorem √584 is the diagonal of a 10 × 22 rectangle — and the distance between the points (0, 0) and (10, 22) on a grid.
• Since √584 = 2√146, a length of √584 is exactly 2 copies of the length √146 laid end to end.

Square roots near √584 and related roots

• The cube root of 584 is about 8.358678.
• Because 584 = 4 × 146, the root is twice √146: 2 × 12.083046 ≈ 24.166092.

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