Square Root of 188

√188 at a glance

Exact value

2√47

Decimal (10 places)

13.7113092008

Rounded

13.7 · 13.71 · 13.711

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.711309

Prime factorization

2² × 47

Cube root

5.728654

How to simplify √188

Look for the largest perfect square that divides 188. Here it is 4 (2²), because 188 = 4 × 47 and 47 has no square factor left:

The prime factorization tells the same story: 188 = 2² × 47. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 47 stays inside.

Check: (2√47)² = 2² × 47 = 4 × 47 = 188. As a decimal, 2√47 = 2 × 6.8556546004 ≈ 13.7113092008.

Where √188 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √188 lies between 13 and 14. 188 is 19 above 169 and 8 below 196, so the root is closer to 14.

• Straight line between 169 and 196: 13.7037 (0.06% low)
• Tangent from 13, i.e. 13 + 19 ÷ 26: 13.7308 (0.14% high)
• Tangent from 14, i.e. 14 − 8 ÷ 28: 13.7143 (0.02% high)

For √188 the tangent at 14 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 188 is just 8 below 196.

Finding √188 with the Babylonian method

Picture a rectangle with an area of 188 and one side x; the other side must be 188 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √188.

Start from the nearest whole number, 14 (14² = 196):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √188 = 13.7113092008 to every decimal shown.

√188 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √188 the pattern is [13; 1, 2, 2, 6, 2, 2, 1, 26] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √188 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 − 188y² = 1. Its smallest solution in positive whole numbers is x = 4,607, y = 336.

√188 in geometry and everyday measurements

• A square room or garden bed covering 188 square feet measures about 13.71 ft (13 ft 9 in) along each wall.
• 188 is not a sum of two whole-number squares — the prime factor 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √188 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √188 as its space diagonal.
• Since √188 = 2√47, a length of √188 is exactly 2 copies of the length √47 laid end to end.

Square roots near √188 and related roots

• The cube root of 188 is about 5.728654.
• Four times the radicand doubles the root: √752 = 2 × √188 ≈ 27.422618.

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