Question

How do you solve `2x^(3/4)=54`?

Answer 1

`x = 81`

Explanation:

Divide both sides by `2` to get:

`x^(3/4) = 27 = 3^3`

If `x` is Real and positive then so is `x^(3/4)` and we can raise both sides to the power `4/3` to find:

`x = x^1 = x^(3/4*4/3) = (x^(3/4))^(4/3) = (3^3)^(4/3) = 3^(3*4/3) = 3^4 = 81`

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Other solutions?

Are there any negative or Complex solutions?

Suppose `x = r(cos theta + i sin theta)` where `r > 0` and `theta in (-pi, pi]`

Then:

`x^(3/4) = r^(3/4)(cos ((3theta)/4) + i sin ((3 theta)/4))`

For the imaginary part to be `0`, we must have `sin ((3theta)/4) = 0`

So `theta = (4pi)/3 k` for some integer `k`

This can only lie in the range `(-pi, pi]` if `k = 0`, so `theta = 0`.

So the only solution is the one on the positive part of the Real axis, i.e. `81`.