how to solve y^4+y^(2^1/3)-2=0

Question

i thing it will solve by quadratic formula

Answer 1

how to solve y^4+y^(2^1/3)-2=0

There is only one solution, y = 1 (By observation)

2^(1/3) is an irrational number

Therefore y cannot be negative since you cannot take a negative number to an irrational power.

(Think about it: what is the sign of (-1)^pi ?)

Let f(y) = y^4+y^(2^1/3)-2

Then f'(y) = 4y^3 + (2^(1/3)*y^{2^(1/3)-1}

Since y is always positive, then f'(y) will always be positive.

I.e the slope of the curve f(y) is always increasing,

Therefore there will be no turning point.

So the curve can't curve back down and cut the x-axis to give another solution for y.

Conclusion: There is only one solution, y = 1