Question

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

Answer 1

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


`\bf 19^{(7)/(4)+(b)/(a)}=19^{(6)/(2)}\implies 19^{(7)/(4)+(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}`