Simplify [(-9)^17 (-9)^0]/[(-9)^14 (-9)] - [(5^14)^6]/[(5^4)^20 * 5]

[lotus]

Simplify Completely:

\(\displaystyle \dfrac{(-9)^{17}\, (-9)^0}{(-9)^{14}\, (-9)}\, -\, \dfrac{(5^{14})^6}{(5^4)^{20}\, \cdot\, 5}\, =\, ?\)

\(\displaystyle \dfrac{(-9)^{17}\, (1)}{(-9)^{14}\, (-9)}\, -\, \dfrac{(5^{14 \cdot 6})}{(5^{4\cdot 20})\, \cdot\, 5}\)

\(\displaystyle \dfrac{-9^{17}}{-9^{14}}\, -\, \dfrac{5^{84}}{5^{80}\, \cdot \, 5}\)

Confused as to what to do next.

 

[MarkFL]

Well, for starters, you have to be careful, as in general:

\(\displaystyle (-a)^b\ne -a^b\)

The original expression \(\displaystyle E\) is:

\(\displaystyle \displaystyle E=\frac{(-9)^{17}(-9)^{0}}{(-9)^{14}(-9)}-\frac{\left(5^{14}\right)^6}{\left(5^{4}\right)^{20}\cdot5}\)

Using the rules:

\(\displaystyle a^ba^c=a^{b+c}\)

\(\displaystyle \left(a^b\right)^c=a^{bc}\)

We have:

\(\displaystyle \displaystyle E=\frac{(-9)^{17}}{(-9)^{15}}-\frac{5^{84}}{5^{81}}\)

Now, use the rule:

\(\displaystyle \displaystyle \frac{a^b}{a^c}=a^{b-c}\)