Square Root of 157

√157 at a glance

Exact value

√157

Decimal (10 places)

12.5299640861

Rounded

12.5 · 12.53 · 12.530

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.529964

Prime factorization

157

Cube root

5.394691

How to simplify √157

157 is a prime number, so its only factors are 1 and 157. There is no perfect-square factor to pull out, which means √157 is already in its simplest radical form.

The square root of any prime is irrational. If √157 were a fraction a/b in lowest terms, then a² = 157b², so 157 would divide a — and then 157 would divide b too, contradicting “lowest terms.” That is why the decimal 12.5299640861 is only a rounded value.

Where √157 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √157 lies between 12 and 13. 157 is 13 above 144 and 12 below 169, so the root is closer to 13.

• Straight line between 144 and 169: 12.5200 (0.08% low)
• Tangent from 12, i.e. 12 + 13 ÷ 24: 12.5417 (0.09% high)
• Tangent from 13, i.e. 13 − 12 ÷ 26: 12.5385 (0.07% high)

For √157 the tangent at 13 wins, missing by only 0.0085. Tangent estimates shine when the number sits close to a perfect square — here 157 is just 12 below 169.

Finding √157 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 157: following the tangent line down to zero simplifies to averaging x with 157 ÷ x.

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

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √157 = 12.5299640861 to every decimal shown.

√157 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √157 the pattern is [12; 1, 1, 7, 1, 5, 2, 1, 1, 1, 1, 2, 5, …] with the block of 17 terms after the semicolon repeating forever (only the first 12 of the 17 are shown). A pattern that never ends is one more proof that √157 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 − 157y² = 1. Its smallest solution in positive whole numbers is x = 46,698,728,731,849, y = 3,726,964,292,220 — 14 digits for x, even though 157 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 4,832,118² − 157 × 385,645² = −1.

√157 in geometry and everyday measurements

• A square room or garden bed covering 157 square feet measures about 12.53 ft (12 ft 6 in) along each wall.
• 157 = 6² + 11², so by the Pythagorean theorem √157 is the diagonal of a 6 × 11 rectangle — and the distance between the points (0, 0) and (6, 11) on a grid.

Square roots near √157 and related roots

• The cube root of 157 is about 5.394691.
• Four times the radicand doubles the root: √628 = 2 × √157 ≈ 25.059928.

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