Question

if
${x}^{a } = {y}^{b} = {z}^{c} and \: {y}^{2} = zx \: then$

Answer 1

$\textbf{ Hello Mate! }$

${x}^{a} = {y}^{b } = {z}^{c} = k$

Here, k is constant.

${x}^{a} = k = > x = {k}^{ \frac{1}{a} }$

Similarily

$y= {k}^{ \frac{1}{b} } \: and \: z = {k}^{ \frac{1}{c} }$

As given,

${y}^{2} = zx \\ {( {k}^{ \frac{1}{b} }) }^{2} = {k}^{ \frac{1}{c} } \times {k}^{ \frac{1}{a} } \\ {k}^{ \frac{2}{b} } = {k}^{ \frac{1}{c} } \times {k}^{ \frac{1}{a} }$

As bases have k common


$\frac{2}{b} = \frac{1}{c} + \frac{1}{a} \\ \frac{2}{b} = \frac{a + c}{ac} \\ \frac{2ac}{a + c} = b$

NOTE: When common bases multiply then powers add up.

Have great future ahead!