Question

If ${\left(\frac{5}{9}\right)}^4\times {\left(\frac{5}{9}\right)}^{-10}={\left(\frac{5}{9}\right)}^{-4}{\left(\frac{5}{9}\right)}^{2a-1}$, then the value of $a$ is ______ .

• A. $1$
• B. $\frac{-5}{2}$
• C. $\frac{-5}{4}$
• D. $\frac{-1}{2}$

Answer 1

The correct option is D $\frac{-1}{2}$

LHS : ${\left(\frac{5}{9}\right)}^4\times {\left(\frac{5}{9}\right)}^{-10}={\left(\frac{5}{9}\right)}^{4-10}={\left(\frac{5}{9}\right)}^{-6}$

RHS : ${\left(\frac{5}{9}\right)}^{-4}\times {\left(\frac{5}{9}\right)}^{2a-1}={\left(\frac{5}{9}\right)}^{2a-1-4}={\left(\frac{5}{9}\right)}^{2a-5}$

Given that,

LHS $=$ RHS

$\therefore {\left(\frac{5}{9}\right)}^{-6}={\left(\frac{5}{9}\right)}^{2a-5}$

$\therefore -6=2a-5$

$\Rightarrow 2a=5-6$

$\Rightarrow a=\frac{-1}{2}$