Rational Exponents and Radical Equations

x^(1/2) means the square root of x, because x^(1/2) × x^(1/2) = x¹.

In general

x^(m/n) is the nth root of x, raised to the m. The bottom number is the root.

The exponent laws still hold

Nothing new is needed. The same rules for adding and subtracting exponents apply.

Squaring can invent answers

Squaring both sides of an equation can produce solutions that fail the original. Check every one.

A fractional exponent is a root

x^(1/2) is √x, because squaring it gives x¹. In general x^(m/n) is the nth root of x, raised to the m. The notation unifies roots and powers under one set of rules.

The exponent laws still apply

Multiplying adds exponents, dividing subtracts, a power of a power multiplies — all unchanged for fractional exponents. That continuity is exactly why fractional exponents were defined this way.

Squaring can introduce false solutions

Squaring both sides of an equation is not reversible: x = 3 and x = −3 both square to 9. So every solution obtained by squaring must be checked in the original equation, and some will fail.

Isolate the radical first

Get the root alone on one side before squaring. Squaring a sum containing a radical leaves another radical behind, doubling the work. With two radicals, isolate one, square, then isolate the other and square again.