Square Root of 575

√575 at a glance

Exact value

5√23

Decimal (10 places)

23.9791576166

Rounded

24.0 · 23.98 · 23.979

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.979158

Prime factorization

5² × 23

Cube root

8.315517

How to simplify √575

Look for the largest perfect square that divides 575. Here it is 25 (5²), because 575 = 25 × 23 and 23 has no square factor left:

The prime factorization tells the same story: 575 = 5² × 23. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 23 stays inside.

Check: (5√23)² = 5² × 23 = 25 × 23 = 575. As a decimal, 5√23 = 5 × 4.7958315233 ≈ 23.9791576166.

Where √575 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √575 lies between 23 and 24. 575 is 46 above 529 and 1 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.9787 (0% low)
• Tangent from 23, i.e. 23 + 46 ÷ 46: 24.0000 (0.09% high)
• Tangent from 24, i.e. 24 − 1 ÷ 48: 23.9792 (0% high)

For √575 the tangent at 24 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 575 is just 1 below 576.

Finding √575 with the Babylonian method

If a guess is too big, 575 ÷ 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√575) in one step.

Start from the nearest whole number, 24 (24² = 576):

Because the starting guess was already close, two steps are enough to match √575 = 23.9791576166 to every decimal shown.

√575 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √575 the pattern is [23; 1, 46] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √575 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 − 575y² = 1. Its smallest solution in positive whole numbers is x = 24, y = 1.

√575 in geometry and everyday measurements

• A square garage floor of 575 square feet measures about 23.98 ft (24 ft) per side, and its corner-to-corner diagonal is √1150 ≈ 33.9 ft.
• 575 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √575 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √575 as its space diagonal.
• Since √575 = 5√23, a length of √575 is exactly 5 copies of the length √23 laid end to end.

Square roots near √575 and related roots

• The cube root of 575 is about 8.315517.
• Squaring undoes the root: (√575)² = 575, while 575² = 330,625 — the number whose square root is 575.

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