Is 123 a perfect square or can it be simplified into a radical form?

1. Why “perfect square” matters and the quick verdict

A perfect square is an integer whose square root is an integer, so checking whether 123 is a perfect square reduces to asking whether some integer squared equals 123; nothing in the documented lists of perfect squares (e.g., 121 = 11² and 144 = 12²) equals 123, so 123 lies strictly between 121 and 144 and therefore is not a perfect square ^{[4] [5]}.

2. Prime factors and the rule for simplifying radicals

The standard algebraic shortcut for simplifying a square root is to factor the radicand into primes and pull out any repeated (paired) primes; 123 factors as 3 × 41, and neither 3 nor 41 is a perfect square nor appears twice, so there are no square factors to extract — √123 is already in simplest radical form ^{[1] [2]}.

3. Decimal approximation and irrationality

Calculators and multiple educational sources give the principal square root of 123 as about 11.0905365064 (often rounded to 11.0905365 or 11.091 for convenience), and because the decimal expansion does not terminate or repeat it is classified as an irrational number — it cannot be written as a ratio p/q of integers ^{[3] [4] [6]}.

4. Methods to determine and approximate the root: long division and approximation

When a number is non‑perfect, common classroom methods include long division-style square root algorithms or iterative averaging (Newton’s method); these produce successively better decimal approximations for √123 and are the reason some sources list 11.09 or 11.091 as useful rounded values while still noting the underlying irrationality ^{[7] [8] [4]}.

5. Cross-checks and pedagogical context

Multiple independent educational websites and calculators all reach the same two conclusions: 123 is not a perfect square, and √123 cannot be simplified further into a product involving an integer square factor — these concordant cross-checks are visible across examples.com, squareroot.info, and calculator-oriented resources ^{[1] [2] [5]}.

6. Alternatives, approximations, and what “simplified radical” means in practice

Practically speaking, “simplified radical” here means reducing the expression under the radical so that no perfect-square factor remains; since 123 = 3 × 41 has no such factor, the radical expression stays as √123. For applied work one can use the decimal ≈11.0905365 for computations, or the symbol √123 when exactness is required — both representations are standard and explicitly noted in the cited sources ^{[2] [3]}.

Conclusion

The mathematical record is clear and unanimous in the cited reporting: 123 is not a perfect square, √123 is irrational, and the radical √123 is already in its simplest form because 123’s factorization provides no square factors to extract ^{[1] [2] [3]}.