Question

How do you solve `(-2x+6)^(1/5)=(-8+10x)^(1/5)` and find any extraneous solutions?

Answer 1

`7/6`

Explanation:

since the two sides of the original equation are equal, than we can take each side of the original equation to the 5th power, and the results will still be equal. This leads to:

`-2x + 6 = -8 + 10x`

...add 2x to both sides:

`6 = -8 + 12x`

add 8 to both sides:

`14 = 12x`

divide by 12:

`14/12 = x = 7/6`

I don't think there are any extraneous solutions. If the fractional power in the original equation had been "even" (1/2, 1/4, etc), then you'd have to allow for `+-7/16`, but, as it is, I think there's just the one. (Socratic, if I'm wrong, I'd love to learn how!)