Question 1

What is the square root of a number?

Accepted Answer

The square root of a non-negative real number x is the non-negative real number y such that y² = x, written y = √x. Every non-negative real number has exactly one non-negative square root (the 'principal' square root) and one negative square root of equal magnitude: √25 = 5, and the equation y² = 25 has two solutions y = ±5. This convention is standardized in ISO 80000-2:2019 (Quantities and units — Part 2: Mathematics) §2-9.6 and used consistently in the IEEE 754-2019 floating-point standard's sqrt operation. For negative real inputs, √x is undefined over the reals; extending to the complex numbers gives √-1 = i per Euler 1749 (Recherches sur les racines imaginaires des équations), enabling √-4 = 2i.

Question 2

Which numbers have integer square roots?

Accepted Answer

Only the perfect squares — numbers of the form k² for some non-negative integer k — have integer square roots. The perfect squares below 200 are: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196. Every other non-negative integer has an irrational square root (Theodorus of Cyrene c. 400 BCE proved √3, √5, …, √17 are irrational; Aristotle Prior Analytics I.23 records the Pythagorean proof that √2 is irrational). The proof for √2 is a classic reductio ad absurdum: assume √2 = p/q in lowest terms, square to get 2q² = p², so p is even, p = 2r, so 2q² = 4r², so q² = 2r², so q is also even — contradicting the lowest-terms assumption.

Question 3

How do computers compute square roots?

Accepted Answer

Modern CPUs implement sqrt as a hardware primitive returning correctly-rounded IEEE 754-2019 double-precision results. On x86-64 the SSE2 SQRTSD instruction (Intel 64 and IA-32 Architectures Software Developer's Manual Volume 2 §5.1) delivers a correctly-rounded 64-bit result in ~15-20 cycles on modern Intel and AMD parts. On ARM64 the FSQRT instruction (ARM Architecture Reference Manual ARMv8 §C6.2.100) does the same. Software fallbacks and higher-precision libraries (GNU MPFR, IBM Accurate Mathematical Library) use variants of Newton's method (x_{n+1} = 0.5(x_n + a/x_n), Isaac Newton 1665 De methodis fluxionum et serierum infinitarum) which doubles the number of correct digits per iteration — starting from a good seed such as the fast inverse square root approximation, 4-5 iterations suffice for double precision.

Question 4

What is the difference between √x, x^(1/2), and x^0.5?

Accepted Answer

For non-negative real x, all three expressions represent the same principal square root: √x = x^(1/2) = x^0.5. The equivalence follows from the definition of rational exponents extending the integer-exponent identity x^m × x^n = x^(m+n): setting m = n = 1/2 gives x^(1/2) × x^(1/2) = x^1 = x, which is exactly the defining property of √x. IEEE 754-2019 §9.2 and ISO/IEC 9899:2018 (C17) §7.12.7.5 both define sqrt(x) and pow(x, 0.5) with subtle differences at edge cases: sqrt(-0.0) returns -0.0, but pow(-0.0, 0.5) is implementation-defined; sqrt(NaN) returns NaN; sqrt(+∞) returns +∞. For maximum accuracy always prefer sqrt over pow — the specialized instruction is both faster and more precisely rounded.

Question 5

How do I simplify a square root radical by hand?

Accepted Answer

To simplify √n where n is a positive integer, factor n into a product of perfect squares and a square-free remainder: √n = √(a² × b) = a × √b where b has no square factors greater than 1. Example: √72 = √(36 × 2) = 6√2. Example: √200 = √(100 × 2) = 10√2. Example: √48 = √(16 × 3) = 4√3. The technique reduces to prime factorization: write n as a product of primes, group pairs of identical primes under the radical, extract each pair as one factor outside the radical. This is standard content in algebra I aligned with the Common Core Standards (CCSS.MATH.CONTENT.HSN.RN.A.2) and every SAT/ACT math curriculum since the 1960s.

Square Root Calculator