Question

Explain why the square root of a number is defined to be equal to that number to the 1/2 power.

Answer 1

There are two complementary ways to see why taking the square root is the same as raising to the 1/2 power.

1) Algebraic/axiomatic justification (keep exponent laws true)
- For positive integers n,m we define a^n as repeated multiplication and we want the exponent laws a^r a^s = a^{r+s} and (a^r)^s = a^{rs} to continue to hold for more general exponents r,s.
- If we ask for a number b such that (b)^2 = a, and we want b to behave like an exponent: b = a^x for some x, then (a^x)^2 = a must hold. Using the exponent rule (a^x)^2 = a^{2x} we get a^{2x} = a^1, so 2x = 1 and x = 1/2.
- Thus the number that squares to a should be a^{1/2}. This also generalizes to rational exponents: a^{m/n} is defined so that (a^{m/n})^n = a^m, i.e. a^{m/n} is the nth root of a^m.

2) Analytic/logarithmic construction (for positive a)
- Using logs and exponentials define a^r := e^{r ln a}. Then a^{1/2} = e^{(1/2) ln a}, and squaring gives (a^{1/2})^2 = e^{ln a} = a, so a^{1/2} is a square root of a.

One caveat: for real numbers we usually take the principal square root, the nonnegative root, so for a ≥ 0 the symbol √a denotes the nonnegative number a^{1/2}. For negative a there is no real a^{1/2}; in the complex numbers a^{1/2} exists but is multi-valued (two values differing by a sign).