Square Root of 98

√98 at a glance

Exact value

7√2

Decimal (10 places)

9.8994949366

Rounded

9.9 · 9.90 · 9.899

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.899495

Prime factorization

2 × 7²

Cube root

4.610436

How to simplify √98

Look for the largest perfect square that divides 98. Here it is 49 (7²), because 98 = 49 × 2 and 2 has no square factor left:

The prime factorization tells the same story: 98 = 2 × 7². Each pair of equal primes leaves the radical as one factor, so 7 comes out and 2 stays inside.

Check: (7√2)² = 7² × 2 = 49 × 2 = 98. As a decimal, 7√2 = 7 × 1.4142135624 ≈ 9.8994949366.

Where √98 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √98 lies between 9 and 10. 98 is 17 above 81 and 2 below 100, so the root is closer to 10.

• Straight line between 81 and 100: 9.8947 (0.05% low)
• Tangent from 9, i.e. 9 + 17 ÷ 18: 9.9444 (0.45% high)
• Tangent from 10, i.e. 10 − 2 ÷ 20: 9.9000 (0.01% high)

For √98 the tangent at 10 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 98 is just 2 below 100.

Finding √98 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 10 (10² = 100):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √98 = 9.8994949366 to every decimal shown.

√98 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √98 the pattern is [9; 1, 8, 1, 18] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √98 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 − 98y² = 1. Its smallest solution in positive whole numbers is x = 99, y = 10.

√98 in geometry and everyday measurements

• A square room or garden bed covering 98 square feet measures about 9.9 ft (9 ft 11 in) along each wall.
• 98 = 7² + 7², so by the Pythagorean theorem √98 is the diagonal of a 7 × 7 rectangle — and the distance between the points (0, 0) and (7, 7) on a grid.
• Since √98 = 7√2, a length of √98 is exactly 7 copies of the length √2 laid end to end.

Square roots near √98 and related roots

• The cube root of 98 is about 4.610436.
• Four times the radicand doubles the root: √392 = 2 × √98 ≈ 19.79899.

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