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If ${(a{b^{ - 1}})^{2x - 1}} = {(b{a^{ - 1}})^{x - 2}}$ then what is the value of $x$?
A. $1$
B. $2$
C. $3$
D. $4$

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Last updated date: 15th Jun 2024
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Answer
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Hint: Here we need to proceed by simplifying the given equation by opening the bracket as we know that ${(ab)^x} = {a^x}{b^x}$ which means that when we open the bracket the power gets separated to each variable which is inside the bracket and we must also know that when we have two terms in the power then they are multiplied. For example: If we have ${({a^{ - 1}})^x}$then we can write it as ${a^{ - 1(x)}} = {a^{ - x}}$
After simplifying it we can equate the power of both the sides which are left hand side and right hand side to get the value of the required variable.

Complete step-by-step answer:
Here we are given that ${(a{b^{ - 1}})^{2x - 1}} = {(b{a^{ - 1}})^{x - 2}}$ and we need to find the value of $x$
So firstly we can simplify this equation by opening the bracket and analyse the power of each variable $a{\text{ and }}b$ and we must also know that when we have two terms in the power then they are multiplied. For example: If we have ${({a^{ - 1}})^x}$then we can write it as ${a^{ - 1(x)}} = {a^{ - x}}$
So we can write
${(a{b^{ - 1}})^{2x - 1}} = {(b{a^{ - 1}})^{x - 2}}$
$\Rightarrow$ ${a^{2x - 1}}{b^{ - (2x - 1)}} = {b^{x - 2}}{a^{ - 1(x - 2)}}$
$\Rightarrow$ ${a^{2x - 1}}{b^{ - 2x + 1}} = {b^{x - 2}}{a^{ - x + 2}}$$ - - - - (1)$
Now we have got the left and the right hand side in the simplified form and we need to find the value of $x$
So we know that if we are given ${a^x} = {a^y}$ then we can say that $x = y$ by equating the power of the same variable on both sides.
Similarly equating the power of $a$ on both the sides in the equation (1) we get that
$2x - 1 = - x + 2$
Upon solving it we get
$
\Rightarrow 2x + x = 2 + 1 \\
\Rightarrow 3x = 3 \\
\Rightarrow x = 1 \\
 $
Hence we get that $x = 1$
Now again equating the power of $b$ on both the sides we get
$
\Rightarrow - 2x + 1 = x - 2 \\
\Rightarrow - 2x - x = - 2 - 1 \\
\Rightarrow - 3x = - 3 \\
\Rightarrow x = 1 \\
 $
Hence we get that the value of $x = 1$
Hence we can say that option A is correct.

Note: Here in this type of problem we must be aware of the knowledge that when we have the same variable as the base and their power on both the sides are the variables then we can equate the powers to get the value of the required variable. We must also know that when we have two terms in the power then they are multiplied. For example: If we have ${({a^{ - 1}})^x}$ then we can write it as ${a^{ - 1(x)}} = {a^{ - x}}$