Square Root of 741

√741 at a glance

Exact value

√741

Decimal (10 places)

27.2213151776

Rounded

27.2 · 27.22 · 27.221

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.221315

Prime factorization

3 × 13 × 19

Cube root

9.049114

How to simplify √741

The prime factorization of 741 is 3 × 13 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √741 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 741, 3, 13 and 19 appear an odd number of times, so √741 is irrational and 27.2213151776 is a rounded value.

Where √741 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √741 lies between 27 and 28. 741 is 12 above 729 and 43 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.2182 (0.01% low)
• Tangent from 27, i.e. 27 + 12 ÷ 54: 27.2222 (0% high)
• Tangent from 28, i.e. 28 − 43 ÷ 56: 27.2321 (0.04% high)

For √741 the tangent at 27 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 741 is just 12 above 729.

Finding √741 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 741: following the tangent line down to zero simplifies to averaging x with 741 ÷ x.

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

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √741 = 27.2213151776 to every decimal shown.

√741 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √741 the pattern is [27; 4, 1, 1, 13, 18, 13, 1, 1, 4, 54] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √741 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 − 741y² = 1. Its smallest solution in positive whole numbers is x = 7,352,695, y = 270,108.

√741 in geometry and everyday measurements

• 741 square feet is 68.8 m². Laid out as a square — a small house footprint or a lot — it is about 27.22 ft (27 ft 3 in) on a side.
• 741 is not a sum of two whole-number squares — the prime factor 3 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √741 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 8 × 26 box, because 1² + 8² + 26² = 741.

Square roots near √741 and related roots

• The cube root of 741 is about 9.049114.
• Squaring undoes the root: (√741)² = 741, while 741² = 549,081 — the number whose square root is 741.

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