Question

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

Answer 1

According to the question we have :-

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

To prove :-

The equation is equal to 1

Proof :-

$\frac{{x}^{a(b - c)}}{ {x}^{b(a - c)}} \div { (\frac{ {x}^{b} }{ {x}^{a} })}^{c} \\ \\ \\ = \frac{ {x}^{ab - ac} }{ {x}^{ab - bc} }\div { (\frac{ {x}^{b} }{ {x}^{a} })}^{c} \\ \\ \\ =\frac{ {x}^{ab - ac} }{ {x}^{ab - bc} } \times {(\frac{ {x}^{a} }{ {x}^{b} })}^{c} \\ \\ \\ =\frac{ {x}^{ab - ac} }{ {x}^{ab - bc} } \times \frac{ {x}^{ac} }{ {x}^{bc} } \\ \\ \\ \\ = \frac{ {x}^{(ab - ac + ac)} }{ {x}^{(ab - bc + bc)} } \\ \\ \\ = \frac{ {x}^{ab} }{ {x}^{ab} } \\ \\ \\ = {x}^{ab - ab} \\ \\ \\ = {x}^{0} \\ \\ \\ = 1$

Hence,

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

KEY POINTS TO REMEMBER :-

☸️ aⁿ * aⁿ = a²ⁿ

☸️ (aⁿ)ⁿ = aⁿ*ⁿ

☸️ a^0 = 1