Question

Which of the following is the quotient of b and a?

Answer 1

$\bf 19^{\frac{7}{4}}\cdot \sqrt[a]{19^b}=19^{\frac{5}{2}}\sqrt{19}\\\\ -----------------------------\\\\ a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^{ n}} \qquad \qquad \sqrt[{ m}]{a^{ n}}\implies a^{\frac{{ n}}{{ m}}}\\\\ -----------------------------\\\\ thus\qquad 19^{\frac{7}{4}}\cdot 19^{\frac{b}{a}}=19^{\frac{5}{2}}\cdot 19^{\frac{1}{2}}\implies 19^{\frac{7}{4}+\frac{b}{a}}=19^{\frac{5}{2}+\frac{1}{2}}$

$\bf 19^{\frac{7}{4}+\frac{b}{a}}=19^{\frac{6}{2}}\implies 19^{\frac{7}{4}+\frac{b}{a}}=19^3\impliedby \begin{array}{llll} \textit{same base, thus}\\ \textit{exponents must be the same} \end{array} \\\\\\ \cfrac{7}{4}+\cfrac{b}{a}=3\implies \cfrac{b}{a}=3-\cfrac{7}{4}$