Square Root of 128

√128 at a glance

Exact value

8√2

Decimal (10 places)

11.3137084990

Rounded

11.3 · 11.31 · 11.314

Perfect square?

No — between 11² and 12²

Rational?

Irrational

Both square roots

±11.313708

Prime factorization

2⁷

Cube root

5.039684

How to simplify √128

Look for the largest perfect square that divides 128. Here it is 64 (8²), because 128 = 64 × 2 and 2 has no square factor left:

The prime factorization tells the same story: 128 = 2⁷. Each pair of equal primes leaves the radical as one factor, so 2³ comes out and 2 stays inside.

128 has 3 square factors (4, 16 and 64). Starting with a smaller one still works but takes more rounds: √128 = 2√32, and √32 can be simplified again. Using 64 straight away finishes in one step.

Check: (8√2)² = 8² × 2 = 64 × 2 = 128. As a decimal, 8√2 = 8 × 1.4142135624 ≈ 11.3137084990.

Where √128 sits between perfect squares

121 = 11² and 144 = 12² are the nearest perfect squares, so √128 lies between 11 and 12. 128 is 7 above 121 and 16 below 144, so the root is closer to 11.

• Straight line between 121 and 144: 11.3043 (0.08% low)
• Tangent from 11, i.e. 11 + 7 ÷ 22: 11.3182 (0.04% high)
• Tangent from 12, i.e. 12 − 16 ÷ 24: 11.3333 (0.17% high)

For √128 the tangent at 11 wins, missing by only 0.0045. Tangent estimates shine when the number sits close to a perfect square — here 128 is just 7 above 121.

Finding √128 with the Babylonian method

Picture a rectangle with an area of 128 and one side x; the other side must be 128 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √128.

Start from the nearest whole number, 11 (11² = 121):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √128 = 11.3137084990 to every decimal shown.

√128 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √128 the pattern is [11; 3, 5, 3, 22] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √128 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 − 128y² = 1. Its smallest solution in positive whole numbers is x = 577, y = 51.

√128 in geometry and everyday measurements

• A square room or garden bed covering 128 square feet measures about 11.31 ft (11 ft 4 in) along each wall.
• 128 = 8² + 8², so by the Pythagorean theorem √128 is the diagonal of a 8 × 8 rectangle — and the distance between the points (0, 0) and (8, 8) on a grid.
• Since √128 = 8√2, a length of √128 is exactly 8 copies of the length √2 laid end to end.

Square roots near √128 and related roots

• The cube root of 128 is about 5.039684.
• Four times the radicand doubles the root: √512 = 2 × √128 ≈ 22.627417.

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