Question

if a=xy^p-1,b=xy^q-1 and c=xy^r-1,prove that a^q-r b^r-p c^p-q=1​

Answer 1

EXPLANATION.

$\implies a = xy^{p - 1} . - - - - - (1).$

$\implies b = xy^{q - 1} . - - - - - (2).$

$\implies c = xy^{r - 1} . - - - - - (3).$

As we know that,

To prove.

$\implies a^{q - r} b^{r - p} c^{p - q} = 1.$

Put the values of a, b, c in the equation, we get.

$\implies \bigg(xy^{p - 1} \bigg)^{q - r} \bigg(xy^{q - 1} \bigg)^{r - p} \bigg(xy^{r - 1} \bigg)^{p - q}$

$\implies (x)^{q - r} (y)^{(p - 1)(q - r)} (x)^{r - p} (y)^{(q - 1)(r - p)} (x)^{p - q} (y)^{(r - 1)(p - q)}$

$\implies (x)^{q - r} (x)^{r - p} (x)^{p - q} (y)^{(p - 1)(q - r)} (y)^{(q - 1)(r - p)} (y)^{(r - 1)(p - q)}$

$\implies (x)^{(q - r + r - p + p - q)} (y)^{(p - 1)(q - r) + (q - 1)(r - p) + (r - 1)(p - q)}$

$\implies (x)^{0} (y)^{(pq - pr - q + r + qr - pq - r + p + rp - qr - p + q)}$

$\implies (x)^{0} (y)^{0} \ = 1.$

Hence Proved.