#### 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.

**Solution:**

It is given that

a = xy^{p – 1}

Here

a^{q – r} = (xy^{b – 1})^{q – r} = x^{q – r}. y^{(q – r) (p – 1)}

b = xy^{q – 1}

Here

b^{r – p} = (xy^{q – 1})^{r – p} = x^{r – p}. y^{(q – 1) (r – p)}

c = xy^{r – 1}

Here

c^{p – q} = (xy^{r – 1})^{p – q} = x^{p – q}. y^{(r – 1) (p – q)}

Consider

LHS = a^{q – r}. b^{r – p}. c^{p – q}

Substituting the values

= x^{q – r}. y^{(q – r) (p – 1)}. x^{r – p}. y^{(q – 1) (r – p)}. x^{p – q}. y^{(r – 1) (p – q)}

By further calculation

= x^{q – r + r – p – q}. y^{(p – 1) (q – r) + (q – 1) (r – p) + (r – 1) (p – q)}

So we get

= x^{0}. y^{pq – pr – q + r + qr – pr – r + p + rp – qr – p + q}

= x^{0}. y^{0}

= 1 × 1

= 1

= RHS

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