Question

Evaluate :

${\left(\frac{17}{19}\right)}^{18}÷{\left(\frac{17}{19}\right)}^{19}$.

Answer 1

Given; ${\left(\frac{17}{19}\right)}^{18}÷{\left(\frac{17}{19}\right)}^{19}$.

We know that

${\mathrm{a}}^{\mathrm{m}}÷{\mathrm{a}}^{\mathrm{n}}={\mathrm{a}}^{\mathrm{m}-\mathrm{n}}$.

Now,

${\left(\frac{17}{19}\right)}^{18}÷{\left(\frac{17}{19}\right)}^{19}={\left(\frac{17}{19}\right)}^{18-19}\left[\because {\mathrm{a}}^{\mathrm{m}}÷{\mathrm{a}}^{\mathrm{n}}={\mathrm{a}}^{\mathrm{m}-\mathrm{n}}\right]$

$$\begin{gathered} ={\left(\frac{17}{19}\right)}^{-1} \\ ={\left(\frac{19}{17}\right)}^1\left[\because {\left(\frac{\mathrm{a}}{\mathrm{b}}\right)}^{-1}={\left(\frac{\mathrm{b}}{\mathrm{a}}\right)}^1\right] \\ =\frac{19}{17} \end{gathered}$$

Hence, value of ${\left(\frac{17}{19}\right)}^{18}÷{\left(\frac{17}{19}\right)}^{19}$ is $\frac{19}{17}$.