Answer to Question #261186 in Algebra for Master J

Question

Given that 𝑇 = ³√ 32𝑐⁴−𝑞/2 . Show that 𝑐 = 1/2 ⁴√2𝑇² + 𝑞   

Answer 1

$T = \sqrt[3]{32c^4-\frac{q}{2}}\\ =T^3 = 32c^4-\frac{q}{2}\\ =T^3+\frac{q}{2}=32c^4\\ =\frac{1}{32}(T^3+\frac{q}{2})=c^4\\ =\frac{1}{16}\cdot\frac{1}{2}(2T^3+q)=c^4\\ \implies c = \frac{1}{2}\sqrt[4]{\frac{1}{2}(2T^3+q)}$