Question

Express with rational denominator:
$\frac{\sqrt[6]{627}}{3-\sqrt[6]{69}}$

Answer 1

Here let $\sqrt[6]{627}=3^{3.\frac{1}{6}}=3^{\frac{1}{2}}=a$ and $\sqrt[6]{69}=3^{2.\frac{1}{6}}=3^{\frac{1}{3}}=b$

We Know that$p^3-q^3=(p-q)(p^2+pq+q^2)$

To rationalize we need to make denominator of the form $p^3-q^3$, since we have power $\frac{1}{3}$ in denominator,

$=\frac{a}{a^2-b}$

Multiplying both denominator and numerator with $a^4+a^{2}b+b^2$

$=\frac{a^5+a^{3}b+a{b}^2}{a^6-b^3}$

Substituting $a=3^{\frac{1}{2}},b=3^{\frac{1}{3}}$

$=\frac{3^{\frac{3}{2}}+3^{\frac{5}{6}}+3^{\frac{1}{6}}}{8}$