Square Root of 295

√295 at a glance

Exact value

√295

Decimal (10 places)

17.1755640373

Rounded

17.2 · 17.18 · 17.176

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.175564

Prime factorization

5 × 59

Cube root

6.656930

How to simplify √295

The prime factorization of 295 is 5 × 59. Every prime appears only once, so there is no pair to bring outside the radical — √295 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 295, 5 and 59 appear an odd number of times, so √295 is irrational and 17.1755640373 is a rounded value.

Where √295 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √295 lies between 17 and 18. 295 is 6 above 289 and 29 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.1714 (0.02% low)
• Tangent from 17, i.e. 17 + 6 ÷ 34: 17.1765 (0.01% high)
• Tangent from 18, i.e. 18 − 29 ÷ 36: 17.1944 (0.11% high)

For √295 the tangent at 17 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 295 is just 6 above 289.

Finding √295 with the Babylonian method

If a guess is too big, 295 ÷ 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√295) in one step.

Start from the nearest whole number, 17 (17² = 289):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √295 = 17.1755640373 to every decimal shown.

√295 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √295 the pattern is [17; 5, 1, 2, 3, 2, 6, 2, 3, 2, 1, 5, 34] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √295 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 − 295y² = 1. Its smallest solution in positive whole numbers is x = 2,024,999, y = 117,900.

√295 in geometry and everyday measurements

• A square patio or deck of 295 square feet is about 17.18 ft (17 ft 2 in) on each side, so edging all the way around takes 4 × √295 ≈ 68.7 ft.
• 295 is not a sum of two whole-number squares — the prime factor 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √295 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 √295 as its space diagonal.

Square roots near √295 and related roots

• The cube root of 295 is about 6.656930.
• Squaring undoes the root: (√295)² = 295, while 295² = 87,025 — the number whose square root is 295.

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