Square Root of 169

√169 at a glance

Exact value

13

Decimal (10 places)

13

Rounded

13

Perfect square?

Yes — 13²

Rational?

Rational (whole number)

Both square roots

±13

Prime factorization

13²

Cube root

5.528775

How to simplify √169

Write 169 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:

169 has 2 factor pairs (1 × 169, 13 × 13). The pair with two equal numbers, 13 × 13, is what makes it a perfect square — and why it has an odd number of factors.

Every perfect square is also a sum of consecutive odd numbers starting at 1. Here 169 = 1 + 3 + 5 + 7 + … + 25, the first 13 odd numbers.

Where √169 sits between perfect squares

169 is 13². The perfect squares on either side are 144 (12²) and 196 (14²), so the gaps are 25 below and 27 above — consecutive squares always differ by consecutive odd numbers.

Finding √169 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 169: following the tangent line down to zero simplifies to averaging x with 169 ÷ x.

To show the method at work on a perfect square, start one above the answer, at 14:

The count of correct decimals went 1, 4, 10 and all 10 over 4 steps — roughly doubling each time — until the guess matched √169 = 13.0000000000 to every decimal shown.

√169 in geometry and everyday measurements

• A square room or garden bed covering 169 square feet measures exactly 13 ft (13 ft) along each wall.
• 169 = 5² + 12², so 5, 12 and 13 form a Pythagorean triple: a right triangle with legs 5 and 12 has a hypotenuse of exactly 13.

Square roots near √169 and related roots

• The cube root of 169 is about 5.528775.
• Taking the square root twice gives the fourth root: √13 ≈ 3.605551.
• Four times the radicand doubles the root: √676 = 2 × √169 ≈ 26.

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