Factoring and simplifying?

Question

x^(4/5) - 4x^(2/5) - 12 =0

Answer 1

There's an easy way to solve a problem like this. When faced with a complicated variable like this, one can rename the variable so that it is easier to deal with in that form, and then back substitute the original variable name in the final solution. Let's call x^(4/5) u^4 instead. Then obviously u = x^(1/5), because [x^(1/5)]^4 = x^(4/5). Then the second term is 4u². Now we can rewrite the entire equation like this:

u^4 - 4u² -12 = 0.

Now we factor the above equation:

u^4 - 4u² -12 = 0
(u² + 2)(u² - 6) = 0

Now we can set each of the above factors equal to 0 and solve for u:

u² + 2 = 0
u² = -2
u = ± √-2
u = ± √(2)(-1)
u = ± (√2) i

or

u² - 6 = 0
u² = 6
u = ± √6

Now we can back substitute:

u = x^(1/5) = ± (√2) i
[x^(1/5)]^5 = [(√2) i]^5 or [-(√2) i]^5
x = 4√2 (i^5) = 4√2 i
or
x = - 4√2 (i^5) = - 4√2 i

For u = ± √6:

u = x^(1/5) = ± √6
[x^(1/5)]^5 = (√6)^5 or (-√6)^5
x = 36 √6
or
x = -36 √6

Answer 2

1) Do not get thrown by the 4/5 and 2/5 powers -- pretend you are working with an x^2 and an x, and then factor like a quadratic equation.

2) (x^(2/5) - 6) (x^(2/5) + 2)

3) Find the roots: x^(2/5) = 6 and x^(2/5) = -2

Answer 3

(x^2/5 -6)(x^2/5 + 2) = 0
Solving...
x^2/5 -6 = 0 or x^2/5 + 2 = 0 (Solve these separately)
Add 6 to both sides Subtract 2 from both sides
x^2/5 = 6 or x^2/5 = -2
Rise both sides to the fifth power on both equations:
x^2 = 6^5 or x^2 = (-2)^5
Take the square root of both sides to get:
x = 6^(5/2) or x = (-2)^(5/2)
I hope that this makes sense. Have a great day.
Thanks,
Eds

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