Just some days ago I appeared for a maths exam. In that exam there was a question related to Indices which I was not able to solve. After the exam I even tried solving it in the home next 2 days but without success. Please help! The question is:
$ a = x^4, \ b =x^2, \ c = x^8$ And
$ x^a = 4 $ Then
$ x^b + x^c = \ ?$
If you want to check your answer I have given it after checking the answer sheet online but they didn't provide the solution.
Answer: 258
$$x=a^{1/4}.$$ $$\ln x^a=a\ln x=a\frac{\ln a}4=\ln 4,$$ $$a\ln a=4\ln 4.$$ Clearly, $a=4$ is a solution*.
Then, $$b=\sqrt 4=2, c=4^2=16,$$ and $$x^b+x^c=4^{2/4}+4^{16/4}=2+256=258.$$
*$(a\ln a)'=\ln a+1$ is positive for $a>-1$, so $a\ln a$ is monotonous and the real root is unique.