Question

If $ {3^x} - {3^{x - 1}} = 18 $ , then $ {x^x} $ is equal to:
 $
  A.\,\,3 \\
  B.\,\,8 \\
  C.\,\,27 \\
  D.\,\,216 \\
  $

Answer 1

Hint: To find required value of ‘x’ we first simplify to write exponents term as individual and then taking $ {3^x}\,\,as\,\,A $ to form a linear equation and then solving linear equation so formed to find value of A and then using this value of A back in substituting to find value of ‘x’ and so required value of $ {x^x} $ and hence required solution of given problem.

Complete step-by-step answer:
Now, writing $ 27 $ in terms of exponent. We have,
 $ {3^x} = {3^3} $
By exponent law, if two exponents are equal and have the same base then their power will also be equal.
Therefore, from the above equation. We have,
 $ x = 3 $ Given, $ {3^x} - {3^{x - 1}} = 18 $
Here we used the reverse of exponent law.
We know that in exponent law in multiplication of two exponents if bases are the same then their powers get added.
But in this problem we use the reverse concept of above exponent law.
In which powers are separated by writing in terms of multiplication.
Therefore, we have
\[
  {3^x} - {3^{x - 1}} = 18 \\
   \Rightarrow {3^x} - {3^x}{.3^{ - 1}} = 18 \\
   \Rightarrow {3^x} - {3^x}.\dfrac{1}{3} = 18 \\
 \]
Now, taking $ {3^x} = A $ in above equation. We have,
 $
  A - \dfrac{A}{3} = 18 \\
   \Rightarrow \dfrac{{3A - A}}{3} = 18 \\
   \Rightarrow \dfrac{{2A}}{3} = 18 \\
   \Rightarrow 2A = 18 \times 3 \\
   \Rightarrow 2A = 54 \\
   \Rightarrow A = \dfrac{{54}}{2} \\
   \Rightarrow A = 27 \\
  $
Substituting value of A in above. We have,
 $
  {3^x} = A \\
   \Rightarrow {3^x} = 27 \;
  $
$\Rightarrow x=3$
But it is required to find the value of $ {x^x} $ .
Therefore, substituting the value of ‘x’ calculated above. We have,
 $
  {3^3} \\
   = 3 \times 3 \times 3 \\
   = 27 \;
  $
Hence, the required value of $ {x^x} $ is $ 27. $
$\Rightarrow x=3$
Therefore, from above we see that out of given options correct option is (C)
So, the correct answer is “Option C”.

Note: In this type of problem in which there are exponent terms with different powers. We first make an exponent term with same base and same power by using exponents laws and then taking such an exponent as some other variable (say A) and forming an equation in term of assumed variable (A) and then solving an equation to find value of A and hence using it value of given variable and so solution of given problem.