Square Root of 151

√151 at a glance

Exact value

√151

Decimal (10 places)

12.2882057274

Rounded

12.3 · 12.29 · 12.288

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.288206

Prime factorization

151

Cube root

5.325074

How to simplify √151

151 is a prime number, so its only factors are 1 and 151. There is no perfect-square factor to pull out, which means √151 is already in its simplest radical form.

The square root of any prime is irrational. If √151 were a fraction a/b in lowest terms, then a² = 151b², so 151 would divide a — and then 151 would divide b too, contradicting “lowest terms.” That is why the decimal 12.2882057274 is only a rounded value.

Where √151 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √151 lies between 12 and 13. 151 is 7 above 144 and 18 below 169, so the root is closer to 12.

• Straight line between 144 and 169: 12.2800 (0.07% low)
• Tangent from 12, i.e. 12 + 7 ÷ 24: 12.2917 (0.03% high)
• Tangent from 13, i.e. 13 − 18 ÷ 26: 12.3077 (0.16% high)

For √151 the tangent at 12 wins, missing by only 0.0035. Tangent estimates shine when the number sits close to a perfect square — here 151 is just 7 above 144.

Finding √151 with the Babylonian method

If a guess is too big, 151 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√151) in one step.

Start from the nearest whole number, 12 (12² = 144):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √151 = 12.2882057274 to every decimal shown.

√151 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √151 the pattern is [12; 3, 2, 7, 1, 3, 4, 1, 1, 1, 11, 1, 1, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √151 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 − 151y² = 1. Its smallest solution in positive whole numbers is x = 1,728,148,040, y = 140,634,693.

√151 in geometry and everyday measurements

• A square room or garden bed covering 151 square feet measures about 12.29 ft (12 ft 3 in) along each wall.
• 151 is not a sum of two whole-number squares — 151 is itself a prime that is one less than a multiple of 4, which rules that out — so √151 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √151 as its space diagonal.

Square roots near √151 and related roots

• The cube root of 151 is about 5.325074.
• Four times the radicand doubles the root: √604 = 2 × √151 ≈ 24.576411.

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