Rationalizing sixth roots?

Relevant wiki: Taylor Series Approximation

$\begin{aligned} \frac 1{\sqrt[6]{5^6+1}-\sqrt[6]{5^6-1}} & = \frac 1{5{\color{#3D99F6}\sqrt[6]{1+\frac 1{5^6}}}-5{\color{#3D99F6}\sqrt[6]{1+\frac 1{5^6}}}} \quad \quad \small {\color{#3D99F6}\text{By Taylor's series expansion}} \\ & = \frac 1{5 {\color{#3D99F6} \left(1 + \frac 16 \cdot \frac 1{5^6} - \frac 5{72} \cdot \frac 1{5^{12}} + ... \right)} - 5 {\color{#3D99F6} \left(1 - \frac 16 \cdot \frac 1{5^6} - \frac 5{72} \cdot \frac 1{5^{12}} - ... \right)}} \\ & \approx \frac 1{5\left(\frac 26 \cdot \frac 1{5^6} \right)} = 3\times 5^5 = \boxed{9375} \end{aligned}$