Question

find the value of x (2/3)2x+1/3 x (2/3) x+1/2 = (2/3)x-15

Answer 1

Dear Student,

We've been given a equation to find the value of variable "x" via exponential rules.

$\bf{(\dfrac{2}{3})^{2x + \dfrac{1}{2}} \times (\dfrac{2}{3})^{x + \dfrac{1}{2}} = (\dfrac{2}{3})^{x - 15}}$

Now, apply the exponential rule into this current expression that is;

$\boxed{\bf{a^b \times a^c = a^{b + c}}}$

Here,  

$\bf{(\dfrac{2}{3})^{2x + \dfrac{1}{2}} \times (\dfrac{2}{3})^{x + \dfrac{1}{2}} = (\dfrac{2}{3})^{2x + \dfrac{1}{2} + x + \dfrac{1}{2}}$

Therefore,

$\bf{(\dfrac{2}{3})^{2x + \dfrac{1}{2} + x + \dfrac{1}{2}} = (\dfrac{2}{3})^{x - 15}}$

Since, the bases are equal that is,  

$\bf{a^{f(x)} = a^{g(x)}, \: then \: \: f(x) = g(x)}$

$\bf{\therefore \quad 2x + \dfrac{1}{2} + x + \dfrac{1}{2} = x - 15}$

$\bf{\therefore \quad 2x + x + \dfrac{1 + 1}{2} = x - 15}$

$\bf{\therefore \quad 3x + 1 = x - 15}$

Subtract by the value of "1" from both sides

$\bf{1 + 3x - 1 = x - 15 -1}$

$\bf{3x = x - 16}$

Subtract the variable of "x" from both the given sides,

$\bf{3x - x = x - 16 - x}$

$\bf{2x = - 16}$

Divide both the given sides by the value of "2";

$\bf{\dfrac{2x}{2} = \dfrac{- 16}{2}}$

$\boxed{\bf{\underline{\therefore \quad Final \: \: Answer; \: \: x = - 8}}}$

Which is the required answer or the final solution for these types of queries.

Hope this helps you and clears your doubts for obtaining the value for variable "x" !!!!!