Square Root of 173

√173 at a glance

Exact value

√173

Decimal (10 places)

13.1529464380

Rounded

13.2 · 13.15 · 13.153

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.152946

Prime factorization

173

Cube root

5.572055

How to simplify √173

173 is a prime number, so its only factors are 1 and 173. There is no perfect-square factor to pull out, which means √173 is already in its simplest radical form.

The square root of any prime is irrational. If √173 were a fraction a/b in lowest terms, then a² = 173b², so 173 would divide a — and then 173 would divide b too, contradicting “lowest terms.” That is why the decimal 13.1529464380 is only a rounded value.

Where √173 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √173 lies between 13 and 14. 173 is 4 above 169 and 23 below 196, so the root is closer to 13.

• Straight line between 169 and 196: 13.1481 (0.04% low)
• Tangent from 13, i.e. 13 + 4 ÷ 26: 13.1538 (0.01% high)
• Tangent from 14, i.e. 14 − 23 ÷ 28: 13.1786 (0.19% high)

For √173 the tangent at 13 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 173 is just 4 above 169.

Finding √173 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 173: following the tangent line down to zero simplifies to averaging x with 173 ÷ x.

Start from the nearest whole number, 13 (13² = 169):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √173 = 13.1529464380 to every decimal shown.

√173 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √173 the pattern is [13; 6, 1, 1, 6, 26] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √173 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 − 173y² = 1. Its smallest solution in positive whole numbers is x = 2,499,849, y = 190,060. Because the period is odd, the equation with −1 on the right also has a solution: 1,118² − 173 × 85² = −1.

√173 in geometry and everyday measurements

• A square room or garden bed covering 173 square feet measures about 13.15 ft (13 ft 2 in) along each wall.
• 173 = 2² + 13², so by the Pythagorean theorem √173 is the diagonal of a 2 × 13 rectangle — and the distance between the points (0, 0) and (2, 13) on a grid.

Square roots near √173 and related roots

• The cube root of 173 is about 5.572055.
• Four times the radicand doubles the root: √692 = 2 × √173 ≈ 26.305893.

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