[tex]\bf a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^{ n}} \qquad \qquad
\sqrt[{ m}]{a^{ n}}\implies a^{\frac{{ n}}{{ m}}}\\\\
-----------------------------\\\\
7^{\frac{19}{4}}\cdot \sqrt[a]{7^b}=7^{\frac{9}{4}}\cdot \sqrt{7^3}\implies 7^{\frac{19}{4}}\cdot \sqrt[a]{7^b}=7^{\frac{9}{4}}\cdot \sqrt[2]{7^3}
\\\\\\
7^{\cfrac{}{}\frac{19}{4}}\cdot 7^{\cfrac{}{}\frac{b}{a}}=7^{\cfrac{}{}\frac{9}{4}}\cdot 7^{\cfrac{}{}\frac{3}{2}}\implies 7^{\cfrac{}{}\frac{19}{4}+\frac{b}{a}}=7^{\cfrac{}{}\frac{9}{4}+\frac{3}{2}}[/tex]
[tex]\bf \textit{same base, thus, the exponents must be the same}
\\\\\\
\cfrac{19}{4}+\cfrac{b}{a}=\cfrac{9}{4}+\cfrac{3}{2}\implies \cfrac{b}{a}=\cfrac{9}{4}+\cfrac{3}{2}-\cfrac{19}{4}\implies \cfrac{b}{a}=\cfrac{9+6-19}{4}
\\\\\\
\cfrac{b}{a}=\cfrac{-4}{4}\implies \cfrac{b}{a}=\cfrac{-1}{1}\implies
\begin{cases}
b=-1\\
a=1
\end{cases}[/tex]
now... if you multiply the numerator and denominator by some same number, the fraction still holds true, for example -3/3 simplified is just -1/1
or -1,000,000/1,000,000 simplified is also -1/1
so.. .the possible values, can be anything, so long the numerator and denominator maintain that ratio
