Question

{6/5}^x×{5/6}^2x=125/216, find x

Answer 1

The value of x is 3 in $\bold{\left(\frac{6}{5}\right)^{x} \times\left(\frac{5}{6}\right)^{2 x}=\frac{125}{216}}$

Solution:

The power of the $\frac{6}{5} \& \frac{5}{6}$  can be equated to find out the term x  

Now to find the term x we need to equal both the terms like changing $\frac{6}{5} t o \frac{5}{6}$ so that we can easily equate the equation.

$\begin{array}{l}{\left(\frac{6}{5}\right)^{x} \times\left(\frac{5}{6}\right)^{2 x}=\frac{125}{216}} \\ \\{\left(\frac{5}{6}\right)^{-x} \times\left(\frac{5}{6}\right)^{2 x}=\frac{125}{216}}\end{array}$

Taking 5/6 common, we get  

$\left(\frac{5}{6}\right)^{-x+2 x}=\frac{125}{216}=\left[\frac{5}{6}\right]^{3}$

Placing 125 and 215 as respective cube of 5 and 6 we get ,

$\left(\frac{5}{6}\right)^{-x+2 x}=\left[\frac{5}{6}\right]^{3}$

As we can see the identity is same, we can equate the powers, after equating the powers we get

-x + 2x = 3

x = 3

Answer 2

Answer:

$Value \: of \: x = 3$

Step-by-step explanation:

$\left(\frac{6}{5}\right)^{x}\times \left(\frac{5}{6}\right)^{2x}=\frac{125}{216}$

$\implies \left(\frac{5}{6}\right)^{-x}\times \left(\frac{5}{6}\right)^{2x}=\left(\frac{5}{6}\right)^{3}$

/*

$By\: Exponential\: law:\\\boxed{\left(\frac{a}{b}\right)^{n}=\left(\frac{b}{a}\right)^{-n}}$

$\implies \left(\frac{5}{6}\right)^{-x}\times \left(\frac{5}{6}\right)^{2x}=\left(\frac{5}{6}\right)^{3}$

$\implies \left(\frac{5}{6}\right)^{-x+2x}=\left(\frac{5}{6}\right)^{3}$

$\boxed {x^{a}\times x^{b}=x^{a+b}}$

$\implies \left(\frac{5}{6}\right)^{x}=\left(\frac{5}{6}\right)^{3}$

$\boxed {If \: a^{m}=a^{n} \: then \: m = n }$

$\implies x = 3$

Therefore,

$Value \: of \: x = 3$

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