Four jobs in one desk: strip perfect squares out of a root with every factor shown, collect like surds across an expression, expand (a+bβc)(d+eβc) the FOIL way, and rationalise a denominator by its conjugate β exact fraction arithmetic and worked lines from GCSE to A-level.
Type β or sqrt(). An outside number multiplies through: 3β48. A negative inside the root turns imaginary.
Terms joined by + or β, each one of: a plain number, βn, or kβn.
Why conjugates work: (a+bβc)(aβbβc) = aΒ² β bΒ²c β the surd cancels and a whole number survives below the line. If that lands on 0, the denominator was zero to begin with and the fraction never existed.