Question

simplify the following​

Answer 1

ANSWER:

To Simplify:

$\:\:\:\bullet\:\:\:\left(a^{\dfrac{1}{x-y}}\right)^{\dfrac{1}{x-z}}\times\left(a^{\dfrac{1}{y-z}}\right)^{\dfrac{1}{y-x}}\times\left(a^{\dfrac{1}{z-x}}\right)^{\dfrac{1}{z-y}}$

Solution:

We are given that,

$\implies\left(a^{\dfrac{1}{x-y}}\right)^{\dfrac{1}{x-z}}\times\left(a^{\dfrac{1}{y-z}}\right)^{\dfrac{1}{y-x}}\times\left(a^{\dfrac{1}{z-x}}\right)^{\dfrac{1}{z-y}}$

We know that,

$\hookrightarrow (p^q)^r=p^{qr}$

So,

$\implies\left(a^{\dfrac{1}{x-y}}\right)^{\dfrac{1}{x-z}}\times\left(a^{\dfrac{1}{y-z}}\right)^{\dfrac{1}{y-x}}\times\left(a^{\dfrac{1}{z-x}}\right)^{\dfrac{1}{z-y}}$

$\implies\left(a^{\dfrac{1}{x-y}\times\dfrac{1}{x-z}}\right)\times\left(a^{\dfrac{1}{y-z}\times\dfrac{1}{y-x}}\right)\times\left(a^{\dfrac{1}{z-x}\times\dfrac{1}{z-y}}\right)$

So,

$\implies\left(a^{\dfrac{1}{(x-y)(x-z)}}\right)\times\left(a^{\dfrac{1}{(y-z)(y-x)}}\right)\times\left(a^{\dfrac{1}{(z-x)(z-y)}}\right)$

We know that,

$\hookrightarrow p^q\times p^r=p^{q+r}$

So,

$\implies\left(a^{\dfrac{1}{(x-y)(x-z)}}\right)\times\left(a^{\dfrac{1}{(y-z)(y-x)}}\right)\times\left(a^{\dfrac{1}{(z-x)(z-y)}}\right)$

$\implies a^{\dfrac{1}{(x-y)(x-z)}+\dfrac{1}{(y-z)(y-x)}+\dfrac{1}{(z-x)(z-y)}}$

$\implies a^{\dfrac{1}{(x-y)(x-z)}-\dfrac{1}{(y-z)(x-y)}+\dfrac{1}{(x-z)(y-z)}}$

Taking LCM,

$\implies a^{\dfrac{1}{(x-y)(x-z)}-\dfrac{1}{(y-z)(x-y)}+\dfrac{1}{(x-z)(y-z)}}$

$\implies a^{\dfrac{(y-z)-(x-z)+(x-y)}{(x-y)(y-z)(x-z)}}$

$\implies a^{\dfrac{y\!\!\!/\:-z\!\!\!/\:-x\!\!\!/\:+z\!\!\!/\:+x\!\!\!/\:-y\!\!\!/\:}{(x-y)(y-z)(x-z)}}$

$\implies a^{\dfrac{0}{(x-y)(y-z)(x-z)}}$

So,

$\implies a^0$

Hence,

$\implies\bf 1$

Therefore,

$\implies\bf\left(a^{\frac{1}{x-y}}\right)^{\frac{1}{x-z}}\times\left(a^{\frac{1}{y-z}}\right)^{\frac{1}{y-x}}\times\left(a^{\frac{1}{z-x}}\right)^{\frac{1}{z-y}}=1$

Answer 2

Answer:

the value of the given expression is 1.

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Hope it helps!!