Square Root of 296

√296 at a glance

Exact value

2√74

Decimal (10 places)

17.2046505341

Rounded

17.2 · 17.20 · 17.205

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.204651

Prime factorization

2³ × 37

Cube root

6.664444

How to simplify √296

Look for the largest perfect square that divides 296. Here it is 4 (2²), because 296 = 4 × 74 and 74 has no square factor left:

The prime factorization tells the same story: 296 = 2³ × 37. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 37 stays inside.

Check: (2√74)² = 2² × 74 = 4 × 74 = 296. As a decimal, 2√74 = 2 × 8.602325267 ≈ 17.2046505341.

Where √296 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √296 lies between 17 and 18. 296 is 7 above 289 and 28 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.2000 (0.03% low)
• Tangent from 17, i.e. 17 + 7 ÷ 34: 17.2059 (0.01% high)
• Tangent from 18, i.e. 18 − 28 ÷ 36: 17.2222 (0.1% high)

For √296 the tangent at 17 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 296 is just 7 above 289.

Finding √296 with the Babylonian method

Picture a rectangle with an area of 296 and one side x; the other side must be 296 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √296.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √296 = 17.2046505341 to every decimal shown.

√296 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √296 the pattern is [17; 4, 1, 7, 1, 4, 34] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √296 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 − 296y² = 1. Its smallest solution in positive whole numbers is x = 3,699, y = 215.

√296 in geometry and everyday measurements

• A square patio or deck of 296 square feet is about 17.2 ft (17 ft 2 in) on each side, so edging all the way around takes 4 × √296 ≈ 68.8 ft.
• 296 = 10² + 14², so by the Pythagorean theorem √296 is the diagonal of a 10 × 14 rectangle — and the distance between the points (0, 0) and (10, 14) on a grid.
• Since √296 = 2√74, a length of √296 is exactly 2 copies of the length √74 laid end to end.

Square roots near √296 and related roots

• The cube root of 296 is about 6.664444.
• Because 296 = 4 × 74, the root is twice √74: 2 × 8.602325 ≈ 17.204651.

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