Question: Simplify: $\dfrac{p + (pq²)^{\frac{1}{3}} + (p²q)^{\frac{1}{3}}}{p-q} ×$$\left ( 1 - \dfrac{q^{\frac{1}{3}}}{p^{\frac{1}{3}}} \right )$


Solution:
Given,
$= \dfrac{p + (pq²)^{\frac{1}{3}} + (p²q)^{\frac{1}{3}}}{p-q} ×\left ( 1 - \dfrac{q^{\frac{1}{3}}}{p^{\frac{1}{3}}} \right )$

$= \dfrac{p + (pq²)^{\frac{1}{3}} + (p²q)^{\frac{1}{3}}}{p-q} ×\left ( \dfrac{p^{\frac{1}{3}} - q^{\frac{1}{3}}}{p^{\frac{1}{3}}} \right )$

$= \dfrac{p^{-\frac{1}{3}} \{ p + (pq²)^{\frac{1}{3}} + (p²q)^{\frac{1}{3}} \} }{p-q} × \left ( p^{\frac{1}{3}} - q^{\frac{1}{3}} \right )$

$= \dfrac{p^{-\frac{1}{3}} \{ p + p^{\frac{1}{3}}q^{\frac{2}{3}} + p^{\frac{2}{3}}q^{\frac{1}{3}} \} }{p-q} × \left ( p^{\frac{1}{3}} - q^{\frac{1}{3}} \right )$

$= \dfrac{p^{1 - \frac{1}{3}} + p^{\frac{1}{3} -\frac{1}{3}}q^{\frac{2}{3}} +p^{\frac{2}{3} -\frac{1}{3}}q^{\frac{1}{3}}}{p-q} × \left ( p^{\frac{1}{3}} - q^{\frac{1}{3}} \right )$

$= \dfrac{p^{\frac{2}{3}} + q^{\frac{2}{3}} + p^{\frac{1}{3}}q^{\frac{1}{3}}}{p-q} × \left ( p^{\frac{1}{3}} - q^{\frac{1}{3}} \right )$

Take $p^{\frac{1}{3}}$ as $a$ and $q^{\frac{1}{3}}$ as $b$. We get the formula of a³-b³.


$= \dfrac{(p^{\frac{1}{3}})^3 - (q^{\frac{1}{3}})^3}{p-q}$

$= \dfrac{p-q}{p-q}$
$= 1$
= Answer
Related Notes and Solutions:

Here is the website link to the notes of Indices.

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