Question

If $\left(4^3\right)\left(4^6\right)\left(4^{12}\right).......\left(4^{3x}\right)=(0.0625{)}^{-54}$, the value of x is

• A. 7
• B. 8
• C. 9
• D. 10

Answer 1

The correct option is B

8

(b) 8

$\left(4^3\right)\left(4^6\right)\left(4^{12}\right).......\left(4^{3x}\right)=(0.0625{)}^{-54}$

$\Rightarrow 4^{(3+6+9+12+....+3x)}={\left(\frac{625}{10000}\right)}^{-54}$

$\Rightarrow 4^{3(1+2+3+4+.+x)}={\left(\frac{1}{16}\right)}^{-54}$

$\Rightarrow 4^{3\left(\frac{x(x+1)}{2}\right)}=(4^{-1}{)}^{-54}$

Comparing both the sides :

$\Rightarrow 3\left(\frac{x(x+1)}{2}\right)=108$

$\Rightarrow x(x+1)=72$

$\Rightarrow x^2+x-72=0$

$\Rightarrow x^2+9x-8x-72=0$

$\Rightarrow x(x+9)-8(x+9)=0$

$\Rightarrow (x+9)(x-8)=0$

$\Rightarrow x=8,-9$

$\Rightarrow x=8$ [$\because$ x is positive]