Square Root of 949

√949 at a glance

Exact value

√949

Decimal (10 places)

30.8058436015

Rounded

30.8 · 30.81 · 30.806

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.805844

Prime factorization

13 × 73

Cube root

9.827025

How to simplify √949

The prime factorization of 949 is 13 × 73. Every prime appears only once, so there is no pair to bring outside the radical — √949 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 949, 13 and 73 appear an odd number of times, so √949 is irrational and 30.8058436015 is a rounded value.

Where √949 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √949 lies between 30 and 31. 949 is 49 above 900 and 12 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.8033 (0.01% low)
• Tangent from 30, i.e. 30 + 49 ÷ 60: 30.8167 (0.04% high)
• Tangent from 31, i.e. 31 − 12 ÷ 62: 30.8065 (0% high)

For √949 the tangent at 31 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 949 is just 12 below 961.

Finding √949 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 949: following the tangent line down to zero simplifies to averaging x with 949 ÷ x.

Start from the nearest whole number, 31 (31² = 961):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √949 = 30.8058436015 to every decimal shown.

√949 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √949 the pattern is [30; 1, 4, 6, 1, 1, 1, 4, 2, 14, 1, 19, 1, …] with the block of 27 terms after the semicolon repeating forever (only the first 12 of the 27 are shown). A pattern that never ends is one more proof that √949 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 − 949y² = 1. Its smallest solution in positive whole numbers is x = 609,622,436,806,639,069,525,576,201, y = 19,789,181,711,517,243,032,971,740 — 27 digits for x, even though 949 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 17,458,843,558,590² − 949 × 566,738,044,393² = −1.

√949 in geometry and everyday measurements

• 949 square feet is 88.2 m². Laid out as a square — a small house footprint or a lot — it is about 30.81 ft (30 ft 10 in) on a side.
• 949 = 7² + 30² = 18² + 25², so by the Pythagorean theorem √949 is the diagonal of rectangles measuring 7 × 30 and 18 × 25 — and the distance between the points (0, 0) and (7, 30) on a grid.

Square roots near √949 and related roots

• The cube root of 949 is about 9.827025.
• Squaring undoes the root: (√949)² = 949, while 949² = 900,601 — the number whose square root is 949.

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