Question

Real numbers $a$ and $b$ satisfy the equations $3^a={81}^{b+2}$ and ${125}^b=5^{a-3}$ The value of $ab$ is

• A. $17$
• B. $9$
• C. $12$
• D. $60$

Answer 1

Answer: (D)

$60$

Given,$3^a={81}^{b+2}\Rightarrow 3^a=3^{4b+8}$
$\Rightarrow a=4b+8\Rightarrow a-4b=8$ ...(i)
and ${125}^b=5^{a-3}\Rightarrow 5^{3b}=5^{a-3}$
$\Rightarrow 3b=a-3\Rightarrow a-3b=3$ ...(ii)
On adding (i) and (ii), we get
$a-4b=8$
$\underset{–}{\underset{-}{a}\underset{+}{-}3b=\underset{-}{3}}$
$-b=5$
$\therefore b=-5$
On putting $b=-5$ in (i), we get
$a=8+4b=8+4\times (-5)\Rightarrow a=-12$
Now, $ab=(-5)(-12)=60$
Hence, Option D is correct.