Square Root of 75

√75 at a glance

Exact value

5√3

Decimal (10 places)

8.6602540378

Rounded

8.7 · 8.66 · 8.660

Perfect square?

No — between 8² and 9²

Rational?

Irrational

Both square roots

±8.660254

Prime factorization

3 × 5²

Cube root

4.217163

How to simplify √75

Look for the largest perfect square that divides 75. Here it is 25 (5²), because 75 = 25 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 75 = 3 × 5². Each pair of equal primes leaves the radical as one factor, so 5 comes out and 3 stays inside.

Check: (5√3)² = 5² × 3 = 25 × 3 = 75. As a decimal, 5√3 = 5 × 1.7320508076 ≈ 8.6602540378.

Where √75 sits between perfect squares

64 = 8² and 81 = 9² are the nearest perfect squares, so √75 lies between 8 and 9. 75 is 11 above 64 and 6 below 81, so the root is closer to 9.

• Straight line between 64 and 81: 8.6471 (0.15% low)
• Tangent from 8, i.e. 8 + 11 ÷ 16: 8.6875 (0.31% high)
• Tangent from 9, i.e. 9 − 6 ÷ 18: 8.6667 (0.07% high)

For √75 the tangent at 9 wins, missing by only 0.0064. Tangent estimates shine when the number sits close to a perfect square — here 75 is just 6 below 81.

Finding √75 with the Babylonian method

If a guess is too big, 75 ÷ 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√75) in one step.

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 √75 = 8.6602540378 to every decimal shown.

√75 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √75 the pattern is [8; 1, 1, 1, 16] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √75 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 − 75y² = 1. Its smallest solution in positive whole numbers is x = 26, y = 3.

√75 in geometry and everyday measurements

• A square room or garden bed covering 75 square feet measures about 8.66 ft (8 ft 8 in) along each wall.
• 75 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √75 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 7 box, because 1² + 5² + 7² = 75.
• Since √75 = 5√3, a length of √75 is exactly 5 copies of the length √3 laid end to end.

Square roots near √75 and related roots

• The cube root of 75 is about 4.217163.
• Four times the radicand doubles the root: √300 = 2 × √75 ≈ 17.320508.

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