Square Root of 747

√747 at a glance

Exact value

3√83

Decimal (10 places)

27.3313007374

Rounded

27.3 · 27.33 · 27.331

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.331301

Prime factorization

3² × 83

Cube root

9.073473

How to simplify √747

Look for the largest perfect square that divides 747. Here it is 9 (3²), because 747 = 9 × 83 and 83 has no square factor left:

The prime factorization tells the same story: 747 = 3² × 83. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 83 stays inside.

Check: (3√83)² = 3² × 83 = 9 × 83 = 747. As a decimal, 3√83 = 3 × 9.1104335791 ≈ 27.3313007374.

Where √747 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √747 lies between 27 and 28. 747 is 18 above 729 and 37 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.3273 (0.01% low)
• Tangent from 27, i.e. 27 + 18 ÷ 54: 27.3333 (0.01% high)
• Tangent from 28, i.e. 28 − 37 ÷ 56: 27.3393 (0.03% high)

For √747 the tangent at 27 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 747 is just 18 above 729.

Finding √747 with the Babylonian method

If a guess is too big, 747 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√747) in one step.

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 √747 = 27.3313007374 to every decimal shown.

√747 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √747 the pattern is [27; 3, 54] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √747 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 − 747y² = 1. Its smallest solution in positive whole numbers is x = 82, y = 3.

√747 in geometry and everyday measurements

• 747 square feet is 69.4 m². Laid out as a square — a small house footprint or a lot — it is about 27.33 ft (27 ft 4 in) on a side.
• 747 is not a sum of two whole-number squares — the prime factor 83 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √747 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 11 × 25 box, because 1² + 11² + 25² = 747.
• Since √747 = 3√83, a length of √747 is exactly 3 copies of the length √83 laid end to end.

Square roots near √747 and related roots

• The cube root of 747 is about 9.073473.
• Squaring undoes the root: (√747)² = 747, while 747² = 558,009 — the number whose square root is 747.

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