Question

If ${{4}^{2x+2}}=64$, then calculate the value of x.
A.$\dfrac{1}{2}$
B.1
C.$\dfrac{3}{2}$
D.2

Answer 1

Hint: We have to solve the value of x in the given equation ${{4}^{2x+2}}=64$. Now, we can write 64 as ${{4}^{3}}$ and we know that in the equation if both the sides have the same base then we can directly equate the power of L.H.S and R.H.S. Using this property, we can write $2x+2=3$. Solving this equation will give the value of x.

Complete step-by-step answer:
The equation given in the question is:
${{4}^{2x+2}}=64$
We know the property that if the base is the same in L.H.S and R.H.S of the equation then we can directly equate the power of L.H.S and R.H.S. In the below, we are demonstrating this property.
$\begin{align}
  & {{a}^{x}}={{a}^{4}} \\
 & \Rightarrow x=4 \\
\end{align}$
Applying this property in the given equation we can write 64 as ${{4}^{3}}$ we get,
${{4}^{2x+2}}={{4}^{3}}$
As you can see that base is same so we can equate the powers on both the sides of the equation as:
$2x+2=3$
Subtracting 2 on both the sides of the equation we get,
$\begin{align}
  & 2x=3-2 \\
 & \Rightarrow 2x=1 \\
\end{align}$
Dividing 2 on both the sides of the equation we get,
$x=\dfrac{1}{2}$
The above solution of the given equation is giving us the value of $x=\dfrac{1}{2}$.
Hence, the correct option is (a).

Note: The alternative way of solving the above problem is to substitute the value of the options in the given equation and see which option is able to satisfy the given equation.
${{4}^{2x+2}}=64$
Substituting the option (a) which is $x=\dfrac{1}{2}$ in the above equation we get,
$\begin{align}
  & {{4}^{2\left( \dfrac{1}{2} \right)+2}}=64 \\
 & \Rightarrow {{4}^{3}}=64 \\
 & \Rightarrow 64=64 \\
\end{align}$
As L.H.S = R.H.S so option (a) is satisfying the given equation.
Similarly, you can check other options also whether they are satisfying the given equation or not. The option which satisfies the given equation is the correct answer.
One more thing that we should keep in mind is that while solving the single choice correct questions, if you get the first option as the right answer then don’t check other options. This trick will save your time in competitive exams.