Question

If ${4^x} - {4^{x - 1}} = 24$ then ${\left( {2x} \right)^x}$ equals
$\left( a \right){\text{ 25}}$
$\left( b \right){\text{ 125}}$
$\left( c \right){\text{ }}\sqrt 5 $
$\left( d \right){\text{ 25}}\sqrt 5 $

Answer 1

Hint: This type of problem can easily be solved just we have to take each step carefully while solving. Simply take the common part outside and then solve for the rest. And we also know when the bases are the same then their powers will become equal to each other. So by using this we can easily solve this question.
Formula used:
When the bases are the same
If ${a^x} = {a^y}$then
$ \Rightarrow x = y$.
Here, $x,y$ are the powers.

Complete step-by-step answer:
Since we have the equation
${4^x} - {4^{x - 1}} = 24$
Now we will take the ${4^x}$common from the left-hand side and keep the right side the same as it is.
So we get
$ \Rightarrow {4^x}\left( {1 - \dfrac{1}{4}} \right) = 24$
Now on solving the above equation, we get
$ \Rightarrow {4^x}\dfrac{3}{4} = 24$
Taking the constant term RHS, we get
$ \Rightarrow {4^x} = 24 \times \dfrac{4}{3}$
Now on dividing and multiplying on the RHS side, we get
$ \Rightarrow {4^x} = 32$
And in the form of power, it can be written as
$ \Rightarrow {4^x} = {2^5}$
Now we will make the base the same on both sides, we get
$ \Rightarrow {2^2}^x = {2^5}$
Since the bases are the same so by using the formula, which is
If ${a^x} = {a^y}$then
$ \Rightarrow x = y$
So on applying the formula, we get
$ \Rightarrow 2x = 5$
Solving for the value of$x$, we get
$ \Rightarrow x = \dfrac{5}{2}$
Since we have to calculate${\left( {2x} \right)^x}$. So on applying the above value in this, we get
$ \Rightarrow {\left( {2 \times \dfrac{5}{2}} \right)^{\dfrac{5}{2}}}$
So on further solving, we get
$ \Rightarrow {\left( 5 \right)^{\dfrac{5}{2}}}$
So it can also be written as
$ \Rightarrow {5^1} \times {5^1} \times {5^{1/2}}$
On solving, we get
$ \Rightarrow 25\sqrt 5 $
Therefore, the correct option is$\left( d \right)$.

Note: So while solving such type of algebraic equation problem we should have to know the properties and concept while solving the equation. So that the equation gets easily minimized and solved with ease. Also, we should always do step by step solutions to reduce the error.