I understand that a fraction is a square if both the numerator and denominator are both squares. I think I may have the answer but I am unsure of what 4 possibilities the answer for x are

The question I have is:

Solve for x by factoring.

question : x^4 = 16/81

x^4 is a square, 16 is a square and 81 is a square so this is what I did...

x^4 - 16/81 = 0

(x^2 + 4/9) ( x^2 - 4/9) = 0

(x^2 + 4/9) (x + 2/3) (x - 2/3) = 0

So does x = 0, + 2/3, -2/3 and then I'm confused on where the 4th answer for x is... pleae help.

You did great up to the last line:


(x^2 + \frac 4 9) (x +\frac 2 3 (x - \frac 2 3 ) = 0


At this point you know two of the roots are 2/3 and -2/3. But you also guessed that a third root is 0 - why do you think that? If you plug 0 for x into the starting equation you would have 0 = 16/81, which is clearly not right.

Instead you can divide the above equation through by (x+2/3)(x-2/3), to yield:


x^2+\frac 4 9 = 0


Rewritten:


x^2 = \frac {-4}9\\
x = \sqrt {\frac {-4} 9 } = \sqrt {\frac {2^2 (-1)} {3^2}},\\
x = \pm \frac 2 3 i