Radical Expressions

Question

How do you know what power to raise both sides of an equation to if the indexes of the radicals are different? For example, in √x+1=³√x−1, what index would you use?

Answer 1

(x+1)^(1/2) = (x-1)^(1/3)

Least common denominator is 6...

(x+1)^(1/2)^6 = (x-1)^(1/3)^6

(x+1)^3 = (x-1)^2

(x+1) (x^2+2x+1) = x^2 - 2x + 1

x^3 + 3x^2 + 3x + 1 = x^2 - 2x + 1

x^3 + 2x^2 + 5x = 0

x ( x^2 + 2x + 5) = 0

x = 0 or x = [-2 +or- sqrt( 4 - 20)] / 2

x=0 or x = [ -2 +or- sqrt(-16)]/2

x = 0 or x = [ -2 +or- 4i]/2

= -1 +or- 2i