Square Root of 168

√168 at a glance

Exact value

2√42

Decimal (10 places)

12.9614813968

Rounded

13.0 · 12.96 · 12.961

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.961481

Prime factorization

2³ × 3 × 7

Cube root

5.517848

How to simplify √168

Look for the largest perfect square that divides 168. Here it is 4 (2²), because 168 = 4 × 42 and 42 has no square factor left:

The prime factorization tells the same story: 168 = 2³ × 3 × 7. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 3 × 7 stays inside.

Check: (2√42)² = 2² × 42 = 4 × 42 = 168. As a decimal, 2√42 = 2 × 6.4807406984 ≈ 12.9614813968.

Where √168 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √168 lies between 12 and 13. 168 is 24 above 144 and 1 below 169, so the root is closer to 13.

• Straight line between 144 and 169: 12.9600 (0.01% low)
• Tangent from 12, i.e. 12 + 24 ÷ 24: 13.0000 (0.3% high)
• Tangent from 13, i.e. 13 − 1 ÷ 26: 12.9615 (0% high)

For √168 the tangent at 13 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 168 is just 1 below 169.

Finding √168 with the Babylonian method

Picture a rectangle with an area of 168 and one side x; the other side must be 168 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √168.

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

The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √168 = 12.9614813968 to every decimal shown.

√168 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √168 the pattern is [12; 1, 24] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √168 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 − 168y² = 1. Its smallest solution in positive whole numbers is x = 13, y = 1.

√168 in geometry and everyday measurements

• A square room or garden bed covering 168 square feet measures about 12.96 ft (13 ft) along each wall.
• 168 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √168 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 8 × 10 box, because 2² + 8² + 10² = 168.
• Since √168 = 2√42, a length of √168 is exactly 2 copies of the length √42 laid end to end.

Square roots near √168 and related roots

• The cube root of 168 is about 5.517848.
• Four times the radicand doubles the root: √672 = 2 × √168 ≈ 25.922963.

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