Square Root of 52

√52 at a glance

Exact value

2√13

Decimal (10 places)

7.2111025509

Rounded

7.2 · 7.21 · 7.211

Perfect square?

No — between 7² and 8²

Rational?

Irrational

Both square roots

±7.211103

Prime factorization

2² × 13

Cube root

3.732511

How to simplify √52

Look for the largest perfect square that divides 52. Here it is 4 (2²), because 52 = 4 × 13 and 13 has no square factor left:

The prime factorization tells the same story: 52 = 2² × 13. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 13 stays inside.

Check: (2√13)² = 2² × 13 = 4 × 13 = 52. As a decimal, 2√13 = 2 × 3.6055512755 ≈ 7.2111025509.

Where √52 sits between perfect squares

49 = 7² and 64 = 8² are the nearest perfect squares, so √52 lies between 7 and 8. 52 is 3 above 49 and 12 below 64, so the root is closer to 7.

• Straight line between 49 and 64: 7.2000 (0.15% low)
• Tangent from 7, i.e. 7 + 3 ÷ 14: 7.2143 (0.04% high)
• Tangent from 8, i.e. 8 − 12 ÷ 16: 7.2500 (0.54% high)

For √52 the tangent at 7 wins, missing by only 0.0032. Tangent estimates shine when the number sits close to a perfect square — here 52 is just 3 above 49.

Finding √52 with the Babylonian method

Picture a rectangle with an area of 52 and one side x; the other side must be 52 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √52.

Start from the nearest whole number, 7 (7² = 49):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √52 = 7.2111025509 to every decimal shown.

√52 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √52 the pattern is [7; 4, 1, 2, 1, 4, 14] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √52 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 − 52y² = 1. Its smallest solution in positive whole numbers is x = 649, y = 90.

√52 in geometry and everyday measurements

• A square room or garden bed covering 52 square feet measures about 7.21 ft (7 ft 3 in) along each wall.
• 52 = 4² + 6², so by the Pythagorean theorem √52 is the diagonal of a 4 × 6 rectangle — and the distance between the points (0, 0) and (4, 6) on a grid.
• Since √52 = 2√13, a length of √52 is exactly 2 copies of the length √13 laid end to end.

Square roots near √52 and related roots

• The cube root of 52 is about 3.732511.
• Four times the radicand doubles the root: √208 = 2 × √52 ≈ 14.422205.

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