{ ((9^((a+1)/b) =125)),((5^(b/a) =3)) :} then faind the volve of ((25^b )/(2a^2 ))=?


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Question Number 163996 by mathlove last updated on 12/Jan/22
${ ((9^((a+1)/b) =125)),((5^(b/a) =3)) :} then faind the volve of ((25^b )/(2a^2 ))=?$
${\begin{cases} 9^{\frac{a + 1}{b}} = 125 \\ 5^{\frac{b}{a}} = 3 \end{cases}$ $t h e n f a i n d t h e v o l v e o f \frac{25^{b}}{2 a^{2}} = ?$
	Answered by nurtani last updated on 12/Jan/22

	$5^{\frac{b}{a}} = 3 \Rightarrow 3 = 5^{\frac{b}{a}} . . . . . (1)$ $9^{\frac{a + 1}{b}} = 125 \Leftrightarrow {(3^{2})}^{(\frac{a + 1}{b})} = 5^{3} \Leftrightarrow 3^{(\frac{2 a + 2}{b})} = 5^{3} . . . . . . (2)$ $s u b t i t u t i o n (1) & (2)$ ${(5^{\frac{b}{a}})}^{(\frac{2 a + 2}{b})} = 5^{3} \Leftrightarrow 5^{(\frac{2 a + 2}{a})} = 5^{3}$ $\Leftrightarrow \frac{2 a + 2}{a} = 3 \Rightarrow 2 a + 2 = 3 a \Rightarrow a = 2$ $5^{\frac{b}{a}} = 3 \Rightarrow 5^{\frac{b}{2}} = 3 \Rightarrow {(5^{b})}^{\frac{1}{2}} = 3 \Rightarrow 5^{b} = 9$ $∴ \frac{25^{b}}{2 a^{2}} = \frac{{(5^{2})}^{b}}{2 a^{2}} = \frac{{(5^{b})}^{2}}{2 a^{2}} = \frac{9^{2}}{2 {(2)}^{2}} = \frac{81}{8}$
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