Question

If $4^X-4^{X-1}$= 24 then the value of $(2X{)}^{\times }$ is_________

• A. $5\sqrt{5}$
• B. $\sqrt{5}$
• C. $25\sqrt{5}$
• D. 125

Answer 1

The correct option is C $25\sqrt{5}$

Consider the problem

Given, $4^X-4^{X-1}=24$

$\begin{array}{c}\Rightarrow {\left(2\right)}^{2X}-{\left(2\right)}^{2\left(X-1\right)}=24\\ \Rightarrow {\left(2\right)}^{2X}-{\left(2\right)}^{2X-2}=24\\ \Rightarrow {\left(2\right)}^{2X}-\frac{{\left(2\right)}^{2X}}{4}=24\end{array}$

On simplifying we get,

${\left(2\right)}^{2X}=32=2^5$

So, $2X=5$

$\Rightarrow X=\frac{5}{2}$

Now,

$\begin{array}{c}{\left(2X\right)}^X={\left(2\times \frac{5}{2}\right)}^{\frac{5}{2}}\\ ={\left(5\right)}^{\frac{5}{2}}\\ =25{\left(5\right)}^{\frac{1}{2}}\\ =25\sqrt{5}\end{array}$

Hence, Option $C$ is the correct answer.