Square Root of 42

√42 at a glance

Exact value

√42

Decimal (10 places)

6.4807406984

Rounded

6.5 · 6.48 · 6.481

Perfect square?

No — between 6² and 7²

Rational?

Irrational

Both square roots

±6.480741

Prime factorization

2 × 3 × 7

Cube root

3.476027

How to simplify √42

The prime factorization of 42 is 2 × 3 × 7. Every prime appears only once, so there is no pair to bring outside the radical — √42 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 42, 2, 3 and 7 appear an odd number of times, so √42 is irrational and 6.4807406984 is a rounded value.

Where √42 sits between perfect squares

36 = 6² and 49 = 7² are the nearest perfect squares, so √42 lies between 6 and 7. 42 is 6 above 36 and 7 below 49, so the root is closer to 6.

• Straight line between 36 and 49: 6.4615 (0.3% low)
• Tangent from 6, i.e. 6 + 6 ÷ 12: 6.5000 (0.3% high)
• Tangent from 7, i.e. 7 − 7 ÷ 14: 6.5000 (0.3% high)

For √42 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √42 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 6 (6² = 36):

The count of correct decimals went 1, 4, 10 and all 10 over 4 steps — roughly doubling each time — until the guess matched √42 = 6.4807406984 to every decimal shown.

√42 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √42 the pattern is [6; 2, 12] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √42 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 − 42y² = 1. Its smallest solution in positive whole numbers is x = 13, y = 2.

√42 in geometry and everyday measurements

• A square room or garden bed covering 42 square feet measures about 6.48 ft (6 ft 6 in) along each wall.
• 42 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 √42 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 5 box, because 1² + 4² + 5² = 42.

Square roots near √42 and related roots

• The cube root of 42 is about 3.476027.
• Four times the radicand doubles the root: √168 = 2 × √42 ≈ 12.961481.

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