SOLUTION: Solve the equation. cuberoot2^(2-x) = 2^(x^2) This problem is very confusing for me, please help!

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Question 61958This question is from textbook
: Solve the equation.
cuberoot2^(2-x) = 2^(x^2)
This problem is very confusing for me, please help! This question is from textbook

Found 2 solutions by stanbon, funmath:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
cuberoot2^(2-x) = 2^(x^2)
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Raise both sides to the 3rd power to get:
2^(2-x)=2^(3x^2)
The exponents must be equal so:
2-x=3x^2
3x^2+x-2=0
(3x-2)(x+1)=0
x=2/3 or x=-1
Cheers,
Stan H.

Answer by funmath(2933) About Me  (Show Source):
You can put this solution on YOUR website!
Solve the equation.
cuberoot2^(2-x) = 2^(x^2)
%282%5E%282-x%29%29%5E%281%2F3%29=2%5E%28x%5E2%29
2%5E%28%281%2F3%29%282-x%29%29=2%5E%28x%5E2%29 Since the bases are the same, their exponents are equal.
%281%2F3%29%282-x%29=x%5E2
3%281%2F3%29%282-x%29=3x%5E2
2-x=3x%5E2
2-2-x%2Bx=3x%5E2%2Bx-2
0=3x%5E2%2Bx-2 Factor by what ever method you use, I'm using arch,ac,or grouping.
0=3x%5E2%2B3x-2x-2
0=%283x%5E2%2B3x%29%2B%28-2x-2%29
0=3x%28x%2B1%29-2%28x%2B1%29
0=%28x%2B1%29%283x-2%29 Set each parntheses=0.
x+1=0 and 3x-2=0
x+1-1=0-1 and 3x-2+2=0+2
x=-1 and 3x=2
x=-1 and 3x/3=2/3
x=-1 and x=2/3
:
Check:
for x=-1
cubedroot 2%5E%282-%28-1%29%29=2%5E%28%28-1%29%5E2%29
cubed root 2%5E%282%2B1%29=2%5E1
cubed root 2%5E3=2
2=2 x=-1 is valid
for x=2/3
cubed root 2%5E%282-%282%2F3%29%29=2%5E%28%282%2F3%29%5E2%29
cubed root 2%5E%286%2F3-2%2F3%29=2%5E%284%2F9%29
cubed root 2%5E%284%2F3%29=2%5E%284%2F9%29
2%5E%284%2F9%29=2%5E%284%2F9%29 x=2/3 is also valid.
There's another method for solving this, but you have to use a calculator if you take the log of both sides, so don't freak out if this is not the method you were taught.
Happy Calculating!!!