Square Root of 24

√24 at a glance

Exact value

2√6

Decimal (10 places)

4.8989794856

Rounded

4.9 · 4.90 · 4.899

Perfect square?

No — between 4² and 5²

Rational?

Irrational

Both square roots

±4.898979

Prime factorization

2³ × 3

Cube root

2.884499

How to simplify √24

Look for the largest perfect square that divides 24. Here it is 4 (2²), because 24 = 4 × 6 and 6 has no square factor left:

The prime factorization tells the same story: 24 = 2³ × 3. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 3 stays inside.

Check: (2√6)² = 2² × 6 = 4 × 6 = 24. As a decimal, 2√6 = 2 × 2.4494897428 ≈ 4.8989794856.

Where √24 sits between perfect squares

16 = 4² and 25 = 5² are the nearest perfect squares, so √24 lies between 4 and 5. 24 is 8 above 16 and 1 below 25, so the root is closer to 5.

• Straight line between 16 and 25: 4.8889 (0.21% low)
• Tangent from 4, i.e. 4 + 8 ÷ 8: 5.0000 (2.06% high)
• Tangent from 5, i.e. 5 − 1 ÷ 10: 4.9000 (0.02% high)

For √24 the tangent at 5 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 24 is just 1 below 25.

Finding √24 with the Babylonian method

Picture a rectangle with an area of 24 and one side x; the other side must be 24 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √24.

Start from the nearest whole number, 5 (5² = 25):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √24 = 4.8989794856 to every decimal shown.

√24 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √24 the pattern is [4; 1, 8] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √24 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 − 24y² = 1. Its smallest solution in positive whole numbers is x = 5, y = 1.

√24 in geometry and everyday measurements

• A square tile with an area of 24 square inches has sides about 4.899 in long.
• 24 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 √24 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 4 box, because 2² + 2² + 4² = 24.
• Since √24 = 2√6, a length of √24 is exactly 2 copies of the length √6 laid end to end.

Square roots near √24 and related roots

• The cube root of 24 is about 2.884499.
• Four times the radicand doubles the root: √96 = 2 × √24 ≈ 9.797959.

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