Square Root of 875

√875 at a glance

Exact value

5√35

Decimal (10 places)

29.5803989155

Rounded

29.6 · 29.58 · 29.580

Perfect square?

No — between 29² and 30²

Rational?

Irrational

Both square roots

±29.580399

Prime factorization

5³ × 7

Cube root

9.564656

How to simplify √875

Look for the largest perfect square that divides 875. Here it is 25 (5²), because 875 = 25 × 35 and 35 has no square factor left:

The prime factorization tells the same story: 875 = 5³ × 7. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 5 × 7 stays inside.

Check: (5√35)² = 5² × 35 = 25 × 35 = 875. As a decimal, 5√35 = 5 × 5.9160797831 ≈ 29.5803989155.

Where √875 sits between perfect squares

841 = 29² and 900 = 30² are the nearest perfect squares, so √875 lies between 29 and 30. 875 is 34 above 841 and 25 below 900, so the root is closer to 30.

• Straight line between 841 and 900: 29.5763 (0.01% low)
• Tangent from 29, i.e. 29 + 34 ÷ 58: 29.5862 (0.02% high)
• Tangent from 30, i.e. 30 − 25 ÷ 60: 29.5833 (0.01% high)

For √875 the tangent at 30 wins, missing by only 0.0029. Tangent estimates shine when the number sits close to a perfect square — here 875 is just 25 below 900.

Finding √875 with the Babylonian method

If a guess is too big, 875 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√875) in one step.

Start from the nearest whole number, 30 (30² = 900):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √875 = 29.5803989155 to every decimal shown.

√875 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √875 the pattern is [29; 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √875 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 − 875y² = 1. Its smallest solution in positive whole numbers is x = 120,126, y = 4,061.

√875 in geometry and everyday measurements

• 875 square feet is 81.3 m². Laid out as a square — a small house footprint or a lot — it is about 29.58 ft (29 ft 7 in) on a side.
• 875 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √875 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 5 × 29 box, because 3² + 5² + 29² = 875.
• Since √875 = 5√35, a length of √875 is exactly 5 copies of the length √35 laid end to end.

Square roots near √875 and related roots

• The cube root of 875 is about 9.564656.
• Squaring undoes the root: (√875)² = 875, while 875² = 765,625 — the number whose square root is 875.

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