Square Root of 425

√425 at a glance

Exact value

5√17

Decimal (10 places)

20.6155281281

Rounded

20.6 · 20.62 · 20.616

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.615528

Prime factorization

5² × 17

Cube root

7.518473

How to simplify √425

Look for the largest perfect square that divides 425. Here it is 25 (5²), because 425 = 25 × 17 and 17 has no square factor left:

The prime factorization tells the same story: 425 = 5² × 17. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 17 stays inside.

Check: (5√17)² = 5² × 17 = 25 × 17 = 425. As a decimal, 5√17 = 5 × 4.1231056256 ≈ 20.6155281281.

Where √425 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √425 lies between 20 and 21. 425 is 25 above 400 and 16 below 441, so the root is closer to 21.

• Straight line between 400 and 441: 20.6098 (0.03% low)
• Tangent from 20, i.e. 20 + 25 ÷ 40: 20.6250 (0.05% high)
• Tangent from 21, i.e. 21 − 16 ÷ 42: 20.6190 (0.02% high)

For √425 the tangent at 21 wins, missing by only 0.0035. Tangent estimates shine when the number sits close to a perfect square — here 425 is just 16 below 441.

Finding √425 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 425: following the tangent line down to zero simplifies to averaging x with 425 ÷ x.

Start from the nearest whole number, 21 (21² = 441):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √425 = 20.6155281281 to every decimal shown.

√425 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √425 the pattern is [20; 1, 1, 1, 1, 1, 1, 40] with the block of 7 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √425 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 − 425y² = 1. Its smallest solution in positive whole numbers is x = 143,649, y = 6,968. Because the period is odd, the equation with −1 on the right also has a solution: 268² − 425 × 13² = −1.

√425 in geometry and everyday measurements

• A square garage floor of 425 square feet measures about 20.62 ft (20 ft 7 in) per side, and its corner-to-corner diagonal is √850 ≈ 29.2 ft.
• 425 = 5² + 20² = 8² + 19² = 13² + 16², so by the Pythagorean theorem √425 is the diagonal of rectangles measuring 5 × 20, 8 × 19 and 13 × 16 — and the distance between the points (0, 0) and (5, 20) on a grid.
• Since √425 = 5√17, a length of √425 is exactly 5 copies of the length √17 laid end to end.

Square roots near √425 and related roots

• The cube root of 425 is about 7.518473.
• Squaring undoes the root: (√425)² = 425, while 425² = 180,625 — the number whose square root is 425.

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