Question

How do you solve $ {3^{x - 1}} = \dfrac{{27}}{{{3^x}}} $ ?

Answer 1

Hint: Here we are given expression having the unknown term and so to find the value of “x” which is the power of the term $ 3 $ so here we will convert both the sides with the same base and then will compare power to get the required value.

Complete step by step solution:
Take the given expression: $ {3^{x - 1}} = \dfrac{{27}}{{{3^x}}} $
The above equation can be re-written as: $ {3^{x - 1}} = \dfrac{{{3^3}}}{{{3^x}}} $
Now using the property of negative rule, that is if the power from the denominator goes to the numerator then positive power becomes negative and vice-versa.
\[ \Rightarrow {3^{x - 1}} = {3^{3 - x}}\]
When bases are equal then the powers are also equal.
 $ \Rightarrow x - 1 = 3 - x $
Move all constants on the right and the variables on the left hand side of the equation. Remember when you move any term from one side to another then the sign of the term also changes. Positive terms become negative and vice versa.
 $ \Rightarrow x + x = 3 + 1 $
Simplify the above equation –
 $ \Rightarrow 2x = 4 $
Term multiplicative on one side if moved to the opposite side, then it goes to the denominator.
 $ \Rightarrow x = \dfrac{4}{2} $
Find factors for the numerator on the right hand side of the equation.
 $ \Rightarrow x = \dfrac{{2 \times 2}}{2} $
Common factors from the numerator and the denominator cancel each other. Therefore, remove from the numerator and denominator in the above equation.
 $ \Rightarrow x = 2 $
This is the required solution.

Thus the correct answer is x= 2.

Note: Always remember that when we expand the brackets or open the brackets, sign outside the bracket is most important. If there is a positive sign outside the bracket then the values inside the bracket does not change and if there is a negative sign outside the bracket then all the terms inside the bracket changes. Positive terms change to negative and negative term changes to positive.