The answer is \(r= \sqrt[3]{GMT^{2}/4 \pi^{2}}\)
\(T^{2} = \frac{4 \pi^{2}}{GM} r^{3}\)
Move \(\frac{4 \pi^{2} }{GM}\) to the other side of the equation:
\(T^{2} /\frac{4 \pi^{2} }{GM} = r^{3} \\
T^{2} *\frac{GM}{4 \pi^{2} } = r^{3}\)
Rearrange:
\(r^{3} = T^{2} *\frac{GM}{4 \pi^{2} } \\
r^{3}= \frac{T^{2} *GM}{4 \pi^{2} } \\
r^{3}= \frac{GMT^{2}}{4 \pi^{2} } \\
r^{3} = GMT^{2}/4 \pi^{2}\)
Since \(x^{3}= \sqrt[3]{x}\), then
\(r^{3} = GMT^{2}/4 \pi^{2} \\
r= \sqrt[3]{GMT^{2}/4 \pi^{2}}\)
