Square Root of 88

√88 at a glance

Exact value

2√22

Decimal (10 places)

9.3808315196

Rounded

9.4 · 9.38 · 9.381

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.380832

Prime factorization

2³ × 11

Cube root

4.447960

How to simplify √88

Look for the largest perfect square that divides 88. Here it is 4 (2²), because 88 = 4 × 22 and 22 has no square factor left:

The prime factorization tells the same story: 88 = 2³ × 11. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 11 stays inside.

Check: (2√22)² = 2² × 22 = 4 × 22 = 88. As a decimal, 2√22 = 2 × 4.6904157598 ≈ 9.3808315196.

Where √88 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √88 lies between 9 and 10. 88 is 7 above 81 and 12 below 100, so the root is closer to 9.

• Straight line between 81 and 100: 9.3684 (0.13% low)
• Tangent from 9, i.e. 9 + 7 ÷ 18: 9.3889 (0.09% high)
• Tangent from 10, i.e. 10 − 12 ÷ 20: 9.4000 (0.2% high)

For √88 the tangent at 9 wins, missing by only 0.0081. Tangent estimates shine when the number sits close to a perfect square — here 88 is just 7 above 81.

Finding √88 with the Babylonian method

Picture a rectangle with an area of 88 and one side x; the other side must be 88 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √88.

Start from the nearest whole number, 9 (9² = 81):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √88 = 9.3808315196 to every decimal shown.

√88 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √88 the pattern is [9; 2, 1, 1, 1, 2, 18] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √88 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 − 88y² = 1. Its smallest solution in positive whole numbers is x = 197, y = 21.

√88 in geometry and everyday measurements

• A square room or garden bed covering 88 square feet measures about 9.38 ft (9 ft 5 in) along each wall.
• 88 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √88 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 6 × 6 box, because 4² + 6² + 6² = 88.
• Since √88 = 2√22, a length of √88 is exactly 2 copies of the length √22 laid end to end.

Square roots near √88 and related roots

• The cube root of 88 is about 4.447960.
• Four times the radicand doubles the root: √352 = 2 × √88 ≈ 18.761663.

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