Square Root of 97

√97 at a glance

Exact value

√97

Decimal (10 places)

9.8488578018

Rounded

9.8 · 9.85 · 9.849

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.848858

Prime factorization

97

Cube root

4.594701

How to simplify √97

97 is a prime number, so its only factors are 1 and 97. There is no perfect-square factor to pull out, which means √97 is already in its simplest radical form.

The square root of any prime is irrational. If √97 were a fraction a/b in lowest terms, then a² = 97b², so 97 would divide a — and then 97 would divide b too, contradicting “lowest terms.” That is why the decimal 9.8488578018 is only a rounded value.

Where √97 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √97 lies between 9 and 10. 97 is 16 above 81 and 3 below 100, so the root is closer to 10.

• Straight line between 81 and 100: 9.8421 (0.07% low)
• Tangent from 9, i.e. 9 + 16 ÷ 18: 9.8889 (0.41% high)
• Tangent from 10, i.e. 10 − 3 ÷ 20: 9.8500 (0.01% high)

For √97 the tangent at 10 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 97 is just 3 below 100.

Finding √97 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 97: following the tangent line down to zero simplifies to averaging x with 97 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √97 = 9.8488578018 to every decimal shown.

√97 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √97 the pattern is [9; 1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18] with the block of 11 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √97 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 − 97y² = 1. Its smallest solution in positive whole numbers is x = 62,809,633, y = 6,377,352. Because the period is odd, the equation with −1 on the right also has a solution: 5,604² − 97 × 569² = −1.

√97 in geometry and everyday measurements

• A square room or garden bed covering 97 square feet measures about 9.85 ft (9 ft 10 in) along each wall.
• 97 = 4² + 9², so by the Pythagorean theorem √97 is the diagonal of a 4 × 9 rectangle — and the distance between the points (0, 0) and (4, 9) on a grid.

Square roots near √97 and related roots

• The cube root of 97 is about 4.594701.
• Four times the radicand doubles the root: √388 = 2 × √97 ≈ 19.697716.

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