Question

Given the exponential equation:, find a common base and solve for x.

Answer 1

EXPLANATION:

Given;

We are given the exponential equation shown below;

$(\frac{125}{8})^{4x-1}=(\frac{4^2}{25^2})^{x+1}$

Required;

We are required to

(i) Find a common base

(ii) Solve for x

Step by step solution;

To solve this problem we shall start with the following steps;

$[(\frac{5}{2})^3]^{4x-1}=[(\frac{2}{5})^4]^{x+1}$

For the left side of the equation, we can refine by applying the rule of exponents;

$\begin{gathered} Flip\text{ the left side of the equation:} \\ (\frac{2}{5})^{-3} \end{gathered}$

Therefore, we now have;

$[(\frac{2}{5})^{-3}]^{4x-1}=[(\frac{2}{5})^4]^{x+1}$$(\frac{2}{5})^{-12x+3}=(\frac{2}{5})^{4x+4}$

We now have a common base and that means;

$\begin{gathered} If: \\ a^x=a^y \\ Then: \\ x=y \end{gathered}$

Therefore;

$-12x+3=4x+4$$-12x-4x=4-3$$-16x=1$

Divide both sides by -16;

$x=-\frac{1}{16}$

ANSWER:

$x=-\frac{1}{16}$