Square Root of 152

√152 at a glance

Exact value

2√38

Decimal (10 places)

12.3288280059

Rounded

12.3 · 12.33 · 12.329

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.328828

Prime factorization

2³ × 19

Cube root

5.336803

How to simplify √152

Look for the largest perfect square that divides 152. Here it is 4 (2²), because 152 = 4 × 38 and 38 has no square factor left:

The prime factorization tells the same story: 152 = 2³ × 19. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 19 stays inside.

Check: (2√38)² = 2² × 38 = 4 × 38 = 152. As a decimal, 2√38 = 2 × 6.164414003 ≈ 12.3288280059.

Where √152 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √152 lies between 12 and 13. 152 is 8 above 144 and 17 below 169, so the root is closer to 12.

• Straight line between 144 and 169: 12.3200 (0.07% low)
• Tangent from 12, i.e. 12 + 8 ÷ 24: 12.3333 (0.04% high)
• Tangent from 13, i.e. 13 − 17 ÷ 26: 12.3462 (0.14% high)

For √152 the tangent at 12 wins, missing by only 0.0045. Tangent estimates shine when the number sits close to a perfect square — here 152 is just 8 above 144.

Finding √152 with the Babylonian method

Picture a rectangle with an area of 152 and one side x; the other side must be 152 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √152.

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 √152 = 12.3288280059 to every decimal shown.

√152 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √152 the pattern is [12; 3, 24] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √152 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 − 152y² = 1. Its smallest solution in positive whole numbers is x = 37, y = 3.

√152 in geometry and everyday measurements

• A square room or garden bed covering 152 square feet measures about 12.33 ft (12 ft 4 in) along each wall.
• 152 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √152 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 12 box, because 2² + 2² + 12² = 152.
• Since √152 = 2√38, a length of √152 is exactly 2 copies of the length √38 laid end to end.

Square roots near √152 and related roots

• The cube root of 152 is about 5.336803.
• Four times the radicand doubles the root: √608 = 2 × √152 ≈ 24.657656.

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