Square Root of 293

√293 at a glance

Exact value

√293

Decimal (10 places)

17.1172427686

Rounded

17.1 · 17.12 · 17.117

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.117243

Prime factorization

293

Cube root

6.641852

How to simplify √293

293 is a prime number, so its only factors are 1 and 293. There is no perfect-square factor to pull out, which means √293 is already in its simplest radical form.

The square root of any prime is irrational. If √293 were a fraction a/b in lowest terms, then a² = 293b², so 293 would divide a — and then 293 would divide b too, contradicting “lowest terms.” That is why the decimal 17.1172427686 is only a rounded value.

Where √293 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √293 lies between 17 and 18. 293 is 4 above 289 and 31 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.1143 (0.02% low)
• Tangent from 17, i.e. 17 + 4 ÷ 34: 17.1176 (0% high)
• Tangent from 18, i.e. 18 − 31 ÷ 36: 17.1389 (0.13% high)

For √293 the tangent at 17 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 293 is just 4 above 289.

Finding √293 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 293: following the tangent line down to zero simplifies to averaging x with 293 ÷ x.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √293 = 17.1172427686 to every decimal shown.

√293 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √293 the pattern is [17; 8, 1, 1, 8, 34] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √293 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 − 293y² = 1. Its smallest solution in positive whole numbers is x = 12,320,649, y = 719,780. Because the period is odd, the equation with −1 on the right also has a solution: 2,482² − 293 × 145² = −1.

√293 in geometry and everyday measurements

• A square patio or deck of 293 square feet is about 17.12 ft (17 ft 1 in) on each side, so edging all the way around takes 4 × √293 ≈ 68.5 ft.
• 293 = 2² + 17², so by the Pythagorean theorem √293 is the diagonal of a 2 × 17 rectangle — and the distance between the points (0, 0) and (2, 17) on a grid.

Square roots near √293 and related roots

• The cube root of 293 is about 6.641852.
• Squaring undoes the root: (√293)² = 293, while 293² = 85,849 — the number whose square root is 293.

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