Question

\sqrt[3]{a} = \sqrt[y]{a} = \sqrt[z]{c} \\ a. \: if \: x = \: \frac{1}{2} \\ y = \frac{1}{3} \: show \: that \: { \frac{a}{b} }^{ \frac{3}{2} } + { \frac{b}{a} }^{ \frac{2}{3} } = {a}^{ \frac{1}{2} } + {b}^{ \frac{ - 1}{3} } \\ c. \: if \: abc = 1 \: pove \: that \: \frac{1}{ {p}^{ - x} + {p}^{y} + 1 } + \frac{1}{ {p}^{ - y} + {p}^{2} + 1 } + \frac{1}{ {p}^{ - z} + {p}^{3} + 1 } = 1​

Answer 1

$\sqrt[3]{a} = \sqrt[y]{a} = \sqrt[z]{c} \\ a. \: if \: x = \: \frac{1}{2} \\ y = \frac{1}{3} \: show \: that \: { \frac{a}{b} }^{ \frac{3}{2} } + { \frac{b}{a} }^{ \frac{2}{3} } = {a}^{ \frac{1}{2} } + {b}^{ \frac{ - 1}{3} } \\ c. \: if \: abc = 1 \: pove \: that \: \frac{1}{ {p}^{ - x} + {p}^{y} + 1 } + \frac{1}{ {p}^{ - y} + {p}^{2} + 1 } + \frac{1}{ {p}^{ - z} + {p}^{3} + 1 } = 1$