If (ab-1)2x-1=(ba-1)x-2 then what is the value of x?

# If ${\left(}{{{\mathit{ab}}}^{{-}{1}}{\right)}}^{{2}{x}{-}{1}}{=}{{{\left(}{\mathit{ba}}}^{{-}{1}}{\right)}}^{{x}{-}{2}}$ then what is the value of x?

1. A
1
2. B
2
3. C
3
4. D
4

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### Solution:

Here we are given that ${\left(}{{{\mathit{ab}}}^{{-}{1}}{\right)}}^{{2}{x}{-}{1}}{=}{{{\left(}{\mathit{ba}}}^{{-}{1}}{\right)}}^{{x}{-}{2}}$ 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 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 ${\left(}{{{a}}^{{-}{1}}{\right)}}^{{x}}$then we can write it as ${\left(}{{{a}}^{{-}{1}}{\right)}}^{{x}}{=}{{a}}^{{-}{x}}$
So we can write
${\left(}{{{\mathit{ab}}}^{{-}{1}}{\right)}}^{{2}{x}{-}{1}}{=}{{{\left(}{\mathit{ba}}}^{{-}{1}}{\right)}}^{{x}{-}{2}}$
${{a}}^{{2}{x}{-}{1}}{{b}}^{{-}{\left(}{2}{x}{-}{1}{\right)}}{=}{{b}}^{{x}{-}{2}}{{a}}^{{-}{1}{\left(}{x}{-}{2}{\right)}}$
${{a}}^{{2}{x}{-}{1}}{{b}}^{{-}{2}{x}{+}{1}}{=}{{b}}^{{x}{-}{2}}{{a}}^{{-}{x}{+}{2}}$−−−−(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 aa on both the sides in the equation (1) we get that
2x−1=−x+2
Upon solving it we get
2x+x=2+1
3x=3
x=1
Hence we get that x=1
Now again equating the power of bb on both the sides we get
2x+1=x2
2xx=21
3x=3

x=1
Hence we get that the value of x=1
Hence we can say that option 1 is correct.

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