Square Root of 783

√783 at a glance

Exact value

3√87

Decimal (10 places)

27.9821371593

Rounded

28.0 · 27.98 · 27.982

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.982137

Prime factorization

3³ × 29

Cube root

9.216950

How to simplify √783

Look for the largest perfect square that divides 783. Here it is 9 (3²), because 783 = 9 × 87 and 87 has no square factor left:

The prime factorization tells the same story: 783 = 3³ × 29. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 3 × 29 stays inside.

Check: (3√87)² = 3² × 87 = 9 × 87 = 783. As a decimal, 3√87 = 3 × 9.3273790531 ≈ 27.9821371593.

Where √783 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √783 lies between 27 and 28. 783 is 54 above 729 and 1 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.9818 (0% low)
• Tangent from 27, i.e. 27 + 54 ÷ 54: 28.0000 (0.06% high)
• Tangent from 28, i.e. 28 − 1 ÷ 56: 27.9821 (0% high)

For √783 the tangent at 28 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 783 is just 1 below 784.

Finding √783 with the Babylonian method

If a guess is too big, 783 ÷ 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√783) in one step.

Start from the nearest whole number, 28 (28² = 784):

Because the starting guess was already close, two steps are enough to match √783 = 27.9821371593 to every decimal shown.

√783 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √783 the pattern is [27; 1, 54] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √783 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 − 783y² = 1. Its smallest solution in positive whole numbers is x = 28, y = 1.

√783 in geometry and everyday measurements

• 783 square feet is 72.7 m². Laid out as a square — a small house footprint or a lot — it is about 27.98 ft (28 ft) on a side.
• 783 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √783 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √783 as its space diagonal.
• Since √783 = 3√87, a length of √783 is exactly 3 copies of the length √87 laid end to end.

Square roots near √783 and related roots

• The cube root of 783 is about 9.216950.
• Squaring undoes the root: (√783)² = 783, while 783² = 613,089 — the number whose square root is 783.

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