Question

[(x^a)/(x^b)]^c×[(x^b)/(x^c)]^c×[(x^c)/(x^a)]^b​

Answer 1

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Step-by-step explanation:

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Answer 2

Step-by-step explanation:

$\bf \underline{Given-}$

$\sf{ \bigg( \frac{ {x}^{a} }{ {x}^{b} } \bigg)^{c} \times\bigg( \frac{ {x}^{b} }{ {x}^{c} } \bigg)^{a} \times \bigg( \frac{ {x}^{c} }{ {x}^{a} } \bigg)^{b} }$

$\bf \underline{To\:find-}$

$\textsf{Simplify the expression and find their value.}$

$\bf \underline{Solution-}$

$\textsf{Given expression}$

$\sf{ \bigg( \frac{ {x}^{a} }{ {x}^{b} } \bigg)^{c} \times\bigg( \frac{ {x}^{b} }{ {x}^{c} } \bigg)^{a} \times \bigg( \frac{ {x}^{c} }{ {x}^{a} } \bigg)^{b} }$

$\sf{\Rightarrow\bigg(x ^{a - b} \bigg)^{c} \times\bigg(x ^{b - c} \bigg)^{a} \times\bigg(x ^{c - a} \bigg)^{b}}$

$\sf{\Rightarrow \: {x}^{ac - bc} \times {x}^{ba - ca} \times {x}^{cb - ab} }$

$\sf{\Rightarrow \: {x}^{ \cancel{ac} - \cancel{bc} + \cancel{ba} - \cancel{ca} + \cancel{cb }- \cancel{ ab}} }$

$\sf{\Rightarrow \: {x}^{0} = 1.}$

$\bf \underline{Answer-}$

$\bf{ \underline{{ Hence, the \: value \: of : \: \bigg( \frac{ {x}^{a} }{ {x}^{b} } \bigg)^{c} \times\bigg( \frac{ {x}^{b} }{ {x}^{c} } \bigg)^{a} \times \bigg( \frac{ {x}^{c} }{ {x}^{a} } \bigg)^{b} \: is \: 1.}}}$

$\bf \underline{More information:-}$

$\textsf{ \: \: \: \: \: \: \:Low of Integral Exponents}$

For any two real numbers a and b, a, b ≠ 0, and for any two positive integers, m and n

➲ If a be any non - zero rational number, then

a^0 = 1

➲ If a be any non - zero rational number and m,n be integer, then

(a^m)^n = a^mn

➲ If a be any non - zero rational number and m be any positive integer, then

a^-m = 1/a^m

➲ If a/b is a rational number and m is a positive integer, then

(a/b)^m = a^m/b^m

➲ For any Integers m and n and any rational number a, a ≠ 0

a^m × a^n = a^m+n

➲ For any Integers m and n for non - zero rational number a,

a^m ÷ a^n = a^m-n

➲ If a and b are non - zero rational numbers and m is any integer, then

(a+b)^m = a^m × b^m

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