Square Root of 748

√748 at a glance

Exact value

2√187

Decimal (10 places)

27.3495886624

Rounded

27.3 · 27.35 · 27.350

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.349589

Prime factorization

2² × 11 × 17

Cube root

9.077520

How to simplify √748

Look for the largest perfect square that divides 748. Here it is 4 (2²), because 748 = 4 × 187 and 187 has no square factor left:

The prime factorization tells the same story: 748 = 2² × 11 × 17. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 11 × 17 stays inside.

Check: (2√187)² = 2² × 187 = 4 × 187 = 748. As a decimal, 2√187 = 2 × 13.6747943312 ≈ 27.3495886624.

Where √748 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √748 lies between 27 and 28. 748 is 19 above 729 and 36 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.3455 (0.02% low)
• Tangent from 27, i.e. 27 + 19 ÷ 54: 27.3519 (0.01% high)
• Tangent from 28, i.e. 28 − 36 ÷ 56: 27.3571 (0.03% high)

For √748 the tangent at 27 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 748 is just 19 above 729.

Finding √748 with the Babylonian method

Picture a rectangle with an area of 748 and one side x; the other side must be 748 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √748.

Start from the nearest whole number, 27 (27² = 729):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √748 = 27.3495886624 to every decimal shown.

√748 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √748 the pattern is [27; 2, 1, 6, 5, 1, 12, 1, 5, 6, 1, 2, 54] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √748 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 − 748y² = 1. Its smallest solution in positive whole numbers is x = 5,658,247, y = 206,886.

√748 in geometry and everyday measurements

• 748 square feet is 69.5 m². Laid out as a square — a small house footprint or a lot — it is about 27.35 ft (27 ft 4 in) on a side.
• 748 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √748 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 6 × 6 × 26 box, because 6² + 6² + 26² = 748.
• Since √748 = 2√187, a length of √748 is exactly 2 copies of the length √187 laid end to end.

Square roots near √748 and related roots

• The cube root of 748 is about 9.077520.
• Because 748 = 4 × 187, the root is twice √187: 2 × 13.674794 ≈ 27.349589.

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