Consider the utility function $U(x,y)=2x^{\frac{3}{2}}y^3$. Determine the indifference curve with $U$-value 54.
$y=\dfrac{9}{\sqrt{x}}$
$y=\dfrac{9}{x^{\frac{3}{2}}}$
$y=\dfrac{3}{x^{\frac{3}{2}}}$
$y=\dfrac{3}{\sqrt{x}}$
Correct:
$$\begin{align}
U(x,y) = 54 & \Leftrightarrow 2x^{\frac{3}{2}}y^3=54\\
& \Leftrightarrow y^3=\frac{54}{2x^{\frac{3}{2}}}\\
& \Leftrightarrow y^3=\frac{27}{x^{\frac{3}{2}}}\\
& \Leftrightarrow y=\Big(\dfrac{27}{x^{\frac{3}{2}}}\Big)^{\frac{1}{3}}\\
& \Leftrightarrow y=\dfrac{3}{x^{\frac{1}{2}}}\\
& \Leftrightarrow y=\frac{3}{\sqrt{x}}.\\
\end{align}$$
Go on.
Wrong: $27^{\frac{1}{3}}\neq 9$.
See Properties power functions.
Wrong: $27^{\frac{1}{3}}\neq 9$.
See Properties power functions.
Wrong: $(x^{\frac{3}{2}})^{\frac{1}{3}}\neq x^{\frac{3}{2}}$.
Try again.