Simplify $\frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}$

Question

How do you make the jump from:

$$\frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}$$

To:

$$\frac{25^{21}-4^{21}}{25^{21}-4(25^{20})}$$

Answer 1

$$ \frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}} = \frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}} \frac{25^{21}}{25^{21}} = \frac{25^{21}-(\frac{4}{25})^{21}25^{21}}{25^{21}-\frac{4}{25}25^{21}} = \frac{25^{21}-4^{21}}{25^{21}-4 \cdot 25^{20}} $$

Answer 2

Have $$\sum_{k=0}^{20}(\frac{4}{25})^k=\frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}=\frac{\frac{1}{25}\cdot(\frac{1}{25})^{20}(25^{21}-4^{21})}{\frac{1}{25}(25-4)}=\frac{25^{21}-4^{21}}{21\cdot25^{20}}$$