Question

show that x^a(b-c)/x ^b(a-c) / (x^b/x^a)^c = 1

Answer 1

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Answer 2

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FORMULA TO BE IMPLEMENTED

We are aware of the below mentioned law of indices

1.

${( {a}^{m} )}^{n} = {a}^{mn}$

2.

$\displaystyle \: \frac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n}$

3.

${a}^{0} = 1$

TO PROVE

$\displaystyle \: \frac{ {x}^{a(b - c)} }{ {x}^{b(a - c)} } \div {\bigg(\frac{ {x}^{b} }{ {x}^{a} } \bigg)}^{c} = 1$

PROOF

$\displaystyle \: \frac{ {x}^{a(b - c)} }{ {x}^{b(a - c)} } \div {\bigg(\frac{ {x}^{b} }{ {x}^{a} } \bigg)}^{c}$

$= \displaystyle \: \frac{ {x}^{(ab - ac)} }{ {x}^{(ab - bc)} } \div {\bigg({ {x}^{b - a} }{ } \bigg)}^{c}$

$= \displaystyle \: { {x}^{(ab - ac - ab + bc)} } \div { {x}^{(bc - ac)} }$

$= \displaystyle \: { {x}^{(bc - ac )} } \div { {x}^{(bc - ac)} }$

$= \displaystyle \: { {x}^{(bc - ac - bc + ac )} }$

$= {x}^{0}$

$= 1$

Hence proved