The value of $$x$$ satisfying $$4^x - 3^{x - 1/2} = 3^{x + 1/2} - 2^{2x - 1}$$ is of the form $$\frac{a}{b}$$ where $$\gcd(a, b) = 1$$. Then the value of $$\left(\frac{a+b}{a-b}\right)$$ is equal to
Solution
Collecting like bases, $$2^{2x} + 2^{2x - 1} = 3^{x + 1/2} + 3^{x - 1/2}$$, that is $$\frac{3}{2} \cdot 2^{2x} = \frac{4}{\sqrt{3}} \cdot 3^{x}$$. This simplifies to $$2^{2x - 3} = 3^{\frac{2x-3}{2}}$$, so $$\left(\frac{2}{\sqrt{3}}\right)^{2x - 3} = 1$$ and $$2x - 3 = 0$$, giving $$x = \frac{3}{2}$$. With $$a = 3$$ and $$b = 2$$, $$\frac{a+b}{a-b} = \frac{5}{1} = 5$$.
Get AI Help
Book Free CAT Mentorship
Get personalized CAT strategy from a 99%iler
500+ students mentored
OTP Verification
Enter the 6-digit code sent to your phone
Booking Summary
Enter OTP
Didn't receive the OTP?
