<s>[INST] Easiest way to solve the equation $x^\frac 43 = \frac {16}{81}$ What would be the easiest way to solve this?
$$x^\frac 43 = \frac {16}{81}$$
I saw this in class and have no clue how did they get $$x = \frac 8{27}$$ 
 [/INST] Given real number $x$ such that $x^\frac{4}{3} = \frac{16}{81}$:
  $\sqrt[3]{x}^4 = (\frac{2}{3})^4$.
  $\sqrt[3]{x} = \frac{2}{3}$ or $\sqrt[3]{x} = -\frac{2}{3}$.
  $x = (\frac{2}{3})^3$ or $x = (-\frac{2}{3})^3 = -(\frac{2}{3})^3$.
To actively counter the common error made by the other respondents, note all the following for real $x$:
  It is false that $(x^4)^\frac{1}{4} = x$. It is true that $(x^4)^\frac{1}{4} = |x|$.
  It is false that $(x^\frac{4}{3})^\frac{3}{4} = x$. It is true that $(x^\frac{4}{3})^\frac{3}{4} = |x|$.
That is the reason why those respondents do not get the two solutions that I have shown above.
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