Square Root of 160

√160 at a glance

Exact value

4√10

Decimal (10 places)

12.6491106407

Rounded

12.6 · 12.65 · 12.649

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.649111

Prime factorization

2⁵ × 5

Cube root

5.428835

How to simplify √160

Look for the largest perfect square that divides 160. Here it is 16 (4²), because 160 = 16 × 10 and 10 has no square factor left:

The prime factorization tells the same story: 160 = 2⁵ × 5. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 2 × 5 stays inside.

160 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √160 = 2√40, and √40 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√10)² = 4² × 10 = 16 × 10 = 160. As a decimal, 4√10 = 4 × 3.1622776602 ≈ 12.6491106407.

Where √160 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √160 lies between 12 and 13. 160 is 16 above 144 and 9 below 169, so the root is closer to 13.

• Straight line between 144 and 169: 12.6400 (0.07% low)
• Tangent from 12, i.e. 12 + 16 ÷ 24: 12.6667 (0.14% high)
• Tangent from 13, i.e. 13 − 9 ÷ 26: 12.6538 (0.04% high)

For √160 the tangent at 13 wins, missing by only 0.0047. Tangent estimates shine when the number sits close to a perfect square — here 160 is just 9 below 169.

Finding √160 with the Babylonian method

Picture a rectangle with an area of 160 and one side x; the other side must be 160 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √160.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √160 = 12.6491106407 to every decimal shown.

√160 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √160 the pattern is [12; 1, 1, 1, 5, 1, 1, 1, 24] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √160 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 − 160y² = 1. Its smallest solution in positive whole numbers is x = 721, y = 57.

√160 in geometry and everyday measurements

• A square room or garden bed covering 160 square feet measures about 12.65 ft (12 ft 8 in) along each wall.
• 160 = 4² + 12², so by the Pythagorean theorem √160 is the diagonal of a 4 × 12 rectangle — and the distance between the points (0, 0) and (4, 12) on a grid.
• Since √160 = 4√10, a length of √160 is exactly 4 copies of the length √10 laid end to end.

Square roots near √160 and related roots

• The cube root of 160 is about 5.428835.
• Four times the radicand doubles the root: √640 = 2 × √160 ≈ 25.298221.

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