Question

If x = (4³) (4⁶) (4⁶) (4⁹) .... (4^3x) = (0.0625)⁻⁵⁴, the value of x is
(a) 7
(b) 8
(c) 9
(d) 10

Answer 1

(b) 8

$$\begin{gathered} \left(4^3\right)\left(4^6\right)\left(4^9\right)\left(4^{12}\right)...\left(4^{3x}\right)={\left(0.0625\right)}^{-54} \\ \Rightarrow 4^{\left(3+6+9+12+...+3\mathrm{x}\right)}={\left(\frac{625}{10000}\right)}^{-54} \\ \Rightarrow 4^{3\left(1+2+3+4+...+x\right)}={\left(\frac{1}{16}\right)}^{-54} \\ \Rightarrow 4^{3\left(\frac{\mathrm{x}\left(\mathrm{x}+1\right)}{2}\right)}={\left(\frac{1}{16}\right)}^{-54} \\ \Rightarrow 4^{3\left(\frac{x\left(x+1\right)}{2}\right)}={\left(4^{-2}\right)}^{-54} \\ \mathrm{Comparing}\mathrm{both}\mathrm{the}\mathrm{sides}: \\ \Rightarrow 3\left(\frac{x\left(x+1\right)}{2}\right)=108 \\ \Rightarrow x\left(x+1\right)=72 \\ \Rightarrow x^2+x-72=0 \\ \Rightarrow x^2+9x-8x-72=0 \\ \Rightarrow x\left(x+9\right)-8\left(x+9\right)=0 \\ \Rightarrow \left(x+9\right)\left(x-8\right)=0 \\ \Rightarrow x=8,-9 \\ \Rightarrow x=8[\because x\mathrm{is}\mathrm{positive}] \end{gathered}$$