Square Root of 553

√553 at a glance

Exact value

√553

Decimal (10 places)

23.5159520326

Rounded

23.5 · 23.52 · 23.516

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.515952

Prime factorization

7 × 79

Cube root

8.208082

How to simplify √553

The prime factorization of 553 is 7 × 79. Every prime appears only once, so there is no pair to bring outside the radical — √553 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 553, 7 and 79 appear an odd number of times, so √553 is irrational and 23.5159520326 is a rounded value.

Where √553 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √553 lies between 23 and 24. 553 is 24 above 529 and 23 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.5106 (0.02% low)
• Tangent from 23, i.e. 23 + 24 ÷ 46: 23.5217 (0.02% high)
• Tangent from 24, i.e. 24 − 23 ÷ 48: 23.5208 (0.02% high)

For √553 the tangent at 24 wins, missing by only 0.0049. Tangent estimates shine when the number sits close to a perfect square — here 553 is just 23 below 576.

Finding √553 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 553: following the tangent line down to zero simplifies to averaging x with 553 ÷ x.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √553 = 23.5159520326 to every decimal shown.

√553 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √553 the pattern is [23; 1, 1, 15, 5, 1, 4, 2, 1, 1, 3, 1, 2, …] with the block of 26 terms after the semicolon repeating forever (only the first 12 of the 26 are shown). A pattern that never ends is one more proof that √553 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 − 553y² = 1. Its smallest solution in positive whole numbers is x = 624,635,837,407, y = 26,562,217,704.

√553 in geometry and everyday measurements

• A square garage floor of 553 square feet measures about 23.52 ft (23 ft 6 in) per side, and its corner-to-corner diagonal is √1106 ≈ 33.3 ft.
• 553 is not a sum of two whole-number squares — the prime factor 7 and 79 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √553 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 15 × 18 box, because 2² + 15² + 18² = 553.

Square roots near √553 and related roots

• The cube root of 553 is about 8.208082.
• Squaring undoes the root: (√553)² = 553, while 553² = 305,809 — the number whose square root is 553.

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