Square Root of 153

√153 at a glance

Exact value

3√17

Decimal (10 places)

12.3693168769

Rounded

12.4 · 12.37 · 12.369

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.369317

Prime factorization

3² × 17

Cube root

5.348481

How to simplify √153

Look for the largest perfect square that divides 153. Here it is 9 (3²), because 153 = 9 × 17 and 17 has no square factor left:

The prime factorization tells the same story: 153 = 3² × 17. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 17 stays inside.

Check: (3√17)² = 3² × 17 = 9 × 17 = 153. As a decimal, 3√17 = 3 × 4.1231056256 ≈ 12.3693168769.

Where √153 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √153 lies between 12 and 13. 153 is 9 above 144 and 16 below 169, so the root is closer to 12.

• Straight line between 144 and 169: 12.3600 (0.08% low)
• Tangent from 12, i.e. 12 + 9 ÷ 24: 12.3750 (0.05% high)
• Tangent from 13, i.e. 13 − 16 ÷ 26: 12.3846 (0.12% high)

For √153 the tangent at 12 wins, missing by only 0.0057. Tangent estimates shine when the number sits close to a perfect square — here 153 is just 9 above 144.

Finding √153 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 153: following the tangent line down to zero simplifies to averaging x with 153 ÷ x.

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

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √153 = 12.3693168769 to every decimal shown.

√153 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √153 the pattern is [12; 2, 1, 2, 2, 2, 1, 2, 24] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √153 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 − 153y² = 1. Its smallest solution in positive whole numbers is x = 2,177, y = 176.

√153 in geometry and everyday measurements

• A square room or garden bed covering 153 square feet measures about 12.37 ft (12 ft 4 in) along each wall.
• 153 = 3² + 12², so by the Pythagorean theorem √153 is the diagonal of a 3 × 12 rectangle — and the distance between the points (0, 0) and (3, 12) on a grid.
• Since √153 = 3√17, a length of √153 is exactly 3 copies of the length √17 laid end to end.

Square roots near √153 and related roots

• The cube root of 153 is about 5.348481.
• Four times the radicand doubles the root: √612 = 2 × √153 ≈ 24.738634.

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