Question

Solution of $ 1 + 3{}^{\dfrac{x}{2}} = {2^x} $ is
 $ A)x = 1 $
 $ B)x = 2 $
 $ C)x = 0 $
 $ D) $ None of these

Answer 1

Hint: First, we need to understand the given question which is in the power values of the digits, and try to find the answer just by applying the given options.
While applying the options we need to find the equivalent condition which is that both the left-hand side and also right-hand side values need to be the same.
We also need to know about the concept of power terms while occurring zero.
Formula used: anything power zero is one, $ {1^0},{2^0},{3^0},.... = 1 $
 $ \sqrt 3 = 1.732 $

Complete step by step answer:
Since from the given that we have, $ 1 + 3{}^{\dfrac{x}{2}} = {2^x} $ and now we need to apply the given options and try to satisfy the equivalent condition that is both the left-hand side and also right-hand sides values need to be same.
Option $ A)x = 1 $ , apply the value of $ x = 1 $ in the given equation $ 1 + 3{}^{\dfrac{x}{2}} = {2^x} $ and then we get, $ 1 + 3{}^{\dfrac{x}{2}} = {2^x} \Rightarrow 1 + 3{}^{\dfrac{1}{2}} = {2^1} $
Since power $ \dfrac{1}{2} $ represents the square root term, then we get $ 1 + 3{}^{\dfrac{1}{2}} = {2^1} \Rightarrow 1 + \sqrt 3 = 2 $
Now applying the value of the root $ 3 $ and thus we get; $ 1 + \sqrt 3 = 2 \Rightarrow 1 + 1.732 = 2 $
By the addition operator we get, $ 1 + 1.732 = 2 \Rightarrow 2.732 \ne 2 $
Thus, right-hand-side values are not equal to the left-hand side, and hence option A is wrong.
Option $ C)x = 0 $ , apply the value of $ x = 0 $ in the given equation $ 1 + 3{}^{\dfrac{x}{2}} = {2^x} $ and then we get, $ 1 + 3{}^{\dfrac{x}{2}} = {2^x} \Rightarrow 1 + 3{}^{\dfrac{0}{2}} = {2^0} $
Since $ \dfrac{0}{2} = 0 $ then we get $ \Rightarrow 1 + {3^0} = {2^0} $
Now apply the power rule for zero that is anything power zero is one, hence we get $ 1 + {3^0} = {2^0} \Rightarrow 1 + 1 = 1 \Rightarrow 2 \ne 1 $
Thus, right-hand-side values are not equal to the left-hand side and hence option C is wrong.
For option $ B)x = 2 $ apply the value of $ x = 2 $ in the given equation $ 1 + 3{}^{\dfrac{x}{2}} = {2^x} $ and then we get,
 $ 1 + 3{}^{\dfrac{x}{2}} = {2^x} \Rightarrow 1 + 3{}^{\dfrac{2}{2}} = {2^2} $ and since $ \dfrac{2}{2} = 1 $ then we get $ \Rightarrow 1 + 3{}^1 = {2^2} $
From the square of number $ 2 $ we have $ {2^2} = 2 \times 2 = 4 $
Thus, we get; $ 1 + 3{}^1 = {2^2} \Rightarrow 1 + 3 = 4 \Rightarrow 4 = 4 $ and hence the right-hand side equals the left-hand side values.

So, the correct answer is “Option B”.

Note: Since we got the one option is correct and thus option $ D) $ None of these is surely wrong.
While solving these kinds of problems, just see the power values and apply the correct power rule to get the correct options, suppose if we apply the power rule $ {2^0} = 2 $ then we get the incorrect solutions.
The addition is the sum or adding the given two or more than two numbers.