Question

If a=xy^p-1, b=yz^q-1,c=zx^r-1, then a^q-r

Answer 1

Answer:

GIVEN :

a = x {y}^{p - 1} \: ,\: b = x {y}^{q - 1} \: , \: c = x {y}^{r- 1}

TO PROVE :

{a}^{q - r} \: {b}^{r - p} \: {c}^{p - q} = 1

PROOF :

LHS

= {a}^{q - r} {b}^{r - p} {c}^{p - q} \\ \\ = {(x {y}^{p - 1} )}^{q - r } \: {(x {y}^{q - 1} )}^{r - p} \: {(x {y}^{r - 1} )}^{p - q} \\ \\ = {x}^{q - r} {y}^{(p - 1)(q - r)} \: {x}^{ r - p} {y}^{(q - 1)(r - p)} \: {x}^{p - q } {y}^{(r - 1)(p - q )} \\ \\ = {x}^{q - r + r - p + p - q} \: {y}^{(p - 1)(q - r) + (q - 1)(r - p) + (r -1 ) (p - q)} \\ \\ = {x}^{0} \: {y}^{pq - pr - q + r + qr - pq - r + p + pr - qr - p + q} \\ \\ = 1 \times {y}^{0} = 1 \times 1 \\ \\ = 1

= RHS -----[ Proved ]------