Question

If we have an expression as $ \sqrt{{{2}^{x-3}}}=16 $ , then the value of $ x $ is
A. $ 5 $
B. $ 4 $
C. $ 8 $
D. $ 11 $

Answer 1

Hint:
First, we will use the inverse function of the square root which is square on both sides of the given equation to convert the given complex equation into the simple form. Now we will get a term which is having a base $ 2 $ in L.H.S. So, we will convert the term which in R.H.S in the form of the base $ 2 $ by factorization. Now we will use the exponential rule and equate the powers of the terms and solve the equation to get the value of $ x $.

Complete step by step answer:
Given that, $ \sqrt{{{2}^{x-3}}}=16 $
Squaring on both sides in the above equation, then we will get
 $ {{\left( \sqrt{{{2}^{x-3}}} \right)}^{2}}={{16}^{2}} $
We know that the square root and square are the inverse functions, so they both are get canceled, then we will have
 $ \Rightarrow {{2}^{x-3}}=256....\left( \text{i} \right) $
In the above equation, we have terms with base $ 2 $ in L.H.S. To proceed further we need to convert the term in R.H.S with a base $ 2 $.
So, consider the term $ 256 $ . Factoring this value
 $ \begin{align}
  & 256=2\times 128 \\
 & \Rightarrow 256=2\times 2\times 64 \\
 & \Rightarrow 256=2\times 2\times 2\times 32 \\
 & \Rightarrow 256=2\times 2\times 2\times 2\times 16 \\
 & \Rightarrow 256=2\times 2\times 2\times 2\times 2\times 8 \\
 & \Rightarrow 256=2\times 2\times 2\times 2\times 2\times 2\times 4 \\
 & \Rightarrow 256=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2 \\
\end{align} $
We can write $ a.a.a.a.a....\text{n times}={{a}^{n}} $ , then we will have
 $ 256={{2}^{8}} $
Substituting the above value in the equation $ \left( \text{i} \right) $ , then we will get
 $ \Rightarrow {{2}^{x-3}}={{2}^{8}} $
We have an exponential rule, that whenever the bases are equal then the powers must be equal. Mathematically if $ {{a}^{m}}={{a}^{n}} $ then must be $ m=n $ . So, from the above equation we can write
 $ \Rightarrow x-3=8 $
Adding $ 3 $ on the both sides of the above equation, then we will have
 $ \Rightarrow x-3+3=8+3 $
We know that $ a-a=0 $ , then we will get
 $ \therefore x=11 $
Hence the value of the $ x $ is $ 11 $ .
$ \therefore $ Option – D is the correct answer.

Note:
 We can check that the obtained answer is correct or not by substituting the obtained $ x $ in the given equation, then we will have
 $ \begin{align}
  & \sqrt{{{2}^{x-3}}}=16 \\
 & \Rightarrow \sqrt{{{2}^{11-3}}}=16 \\
 & \Rightarrow \sqrt{{{2}^{8}}}=16 \\
 & \Rightarrow {{2}^{\dfrac{8}{2}}}=16 \\
 & \Rightarrow {{2}^{4}}=16 \\
 & \Rightarrow 16=16 \\
\end{align} $
So, the obtained answer is correct.