Solve $a=x^n$ , $b=(x+1)^n$ for $x,n$

Question

$$a=x^n~,~b=(x+1)^n$$

Just trying to solve these for $x$ and $n$ . For some reason WolframAlpha gives me a blank screen? Much thanks for any help.

Answer 1

HINT:

$\frac{b}{a}=\left(\frac{x+1}{x}\right)^n$

$\left(\frac{b}{a}\right)^{\frac1n}=1+\frac1x$

$x=\frac{1}{\left(\frac{b}{a}\right)^{\frac1n}-1}$.

Substitute this in $(1)$ to find $n$.

Now use since x is an integer and given $x=\frac{1}{\left(\frac{b}{a}\right)^{\frac1n}-1}$, what can you infer?

EDIT: As x is a positive integer as is said by OP in the comments, hence $\left(\frac{b}{a}\right)^{\frac1n}-1=1$, or $b=a2^n$ or $n=\frac{log{\frac{b}{a}}}{log2}$.

$\therefore n=\frac{log{\frac{b}{a}}}{log2}$ and $x=1$