Square Root of 745

√745 at a glance

Exact value

√745

Decimal (10 places)

27.2946881279

Rounded

27.3 · 27.29 · 27.295

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.294688

Prime factorization

5 × 149

Cube root

9.065368

How to simplify √745

The prime factorization of 745 is 5 × 149. Every prime appears only once, so there is no pair to bring outside the radical — √745 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 745, 5 and 149 appear an odd number of times, so √745 is irrational and 27.2946881279 is a rounded value.

Where √745 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √745 lies between 27 and 28. 745 is 16 above 729 and 39 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.2909 (0.01% low)
• Tangent from 27, i.e. 27 + 16 ÷ 54: 27.2963 (0.01% high)
• Tangent from 28, i.e. 28 − 39 ÷ 56: 27.3036 (0.03% high)

For √745 the tangent at 27 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 745 is just 16 above 729.

Finding √745 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 745: following the tangent line down to zero simplifies to averaging x with 745 ÷ x.

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 √745 = 27.2946881279 to every decimal shown.

√745 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √745 the pattern is [27; 3, 2, 1, 1, 5, 2, 10, 2, 5, 1, 1, 2, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √745 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 − 745y² = 1. Its smallest solution in positive whole numbers is x = 12,769,001, y = 467,820.

√745 in geometry and everyday measurements

• 745 square feet is 69.2 m². Laid out as a square — a small house footprint or a lot — it is about 27.29 ft (27 ft 4 in) on a side.
• 745 = 4² + 27² = 13² + 24², so by the Pythagorean theorem √745 is the diagonal of rectangles measuring 4 × 27 and 13 × 24 — and the distance between the points (0, 0) and (4, 27) on a grid.

Square roots near √745 and related roots

• The cube root of 745 is about 9.065368.
• Squaring undoes the root: (√745)² = 745, while 745² = 555,025 — the number whose square root is 745.

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