Square Root of 536

√536 at a glance

Exact value

2√134

Decimal (10 places)

23.1516738056

Rounded

23.2 · 23.15 · 23.152

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.151674

Prime factorization

2³ × 67

Cube root

8.123096

How to simplify √536

Look for the largest perfect square that divides 536. Here it is 4 (2²), because 536 = 4 × 134 and 134 has no square factor left:

The prime factorization tells the same story: 536 = 2³ × 67. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 67 stays inside.

Check: (2√134)² = 2² × 134 = 4 × 134 = 536. As a decimal, 2√134 = 2 × 11.5758369028 ≈ 23.1516738056.

Where √536 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √536 lies between 23 and 24. 536 is 7 above 529 and 40 below 576, so the root is closer to 23.

• Straight line between 529 and 576: 23.1489 (0.01% low)
• Tangent from 23, i.e. 23 + 7 ÷ 46: 23.1522 (0% high)
• Tangent from 24, i.e. 24 − 40 ÷ 48: 23.1667 (0.06% high)

For √536 the tangent at 23 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 536 is just 7 above 529.

Finding √536 with the Babylonian method

Picture a rectangle with an area of 536 and one side x; the other side must be 536 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √536.

Start from the nearest whole number, 23 (23² = 529):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √536 = 23.1516738056 to every decimal shown.

√536 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √536 the pattern is [23; 6, 1, 1, 2, 5, 2, 1, 1, 6, 46] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √536 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 − 536y² = 1. Its smallest solution in positive whole numbers is x = 145,925, y = 6,303.

√536 in geometry and everyday measurements

• A square garage floor of 536 square feet measures about 23.15 ft (23 ft 2 in) per side, and its corner-to-corner diagonal is √1072 ≈ 32.7 ft.
• 536 is not a sum of two whole-number squares — the prime factor 67 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √536 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 6 × 22 box, because 4² + 6² + 22² = 536.
• Since √536 = 2√134, a length of √536 is exactly 2 copies of the length √134 laid end to end.

Square roots near √536 and related roots

• The cube root of 536 is about 8.123096.
• Because 536 = 4 × 134, the root is twice √134: 2 × 11.575837 ≈ 23.151674.

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