Question

${a}^{x} = b \: \: \: \: {b}^{y} = c \: \: \: \: \: {c}^{z} = a \: \: \: \: \: \: prove \: that \: xyz \: = 1$
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Answer 1

$= > \: \: {a}^{x} + {b}^{y} + {c}^{z} = a + b + c$

On comparing,

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

,

$= > \: \: {b}^{y} = b^{1} \\ \\ = > \: \: y = 1$

and,

$= > \: \: {c}^{z} = c^{1} \\ \\ = > \: \: z = 1$

So,

x = 1; y = 1; z = 1;

=> x × y × z

=> 1 × 1 × 1

=> 1

Hence, Proved.

Answer 2

Given:-

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

To Prove:-

xyz = 1

Solution:-

As,

$\\\: \bold{{a}^{x} = b}$

But Here a = $c^{z}$

$\\ \: \bold{( { {c}^{z} })^{x} = b}$

And Now c = $b^{y}$

$\\ \: \bold{( {( { {b}^{y} })^{z}) }^{x} = b}$

Now, $\;\Large{b^{yzx} = b^{1}}$

As The Bases are equal we can equate powers.

$\;\Large{\sf{\pink{xyz = 1}}}$

Hence Proved, xyz = 1