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# Real numbers problem – C.M.I – U.G – 2019

## Understand the problem

.Find all real numbers x for which
$\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$

##### Source of the problem

C.M.I (Chennai mathematical institute ) U.G-2019

### Real Numbers

8 out of 10

##### Suggested Book

Do you really need a hint? Try it first!

It is of the the form of  $\frac{a^3+b^3}{a^2b+b^2a}$ . Do you observe ?  where a=$2^x$ b=$3^x$

$\frac{a^3+b^3+a^2b+b^2a}{a^2b+b^2a}$=$\frac{13}{6}$ $\Rightarrow\frac{a^2+b^2}{ab}=\frac{13}{6}$ $\frac{a}{b}+\frac{b}{a}=\frac{13}{6}$ let x=a/b then it is a quadratic equation

$6x^2-13x+6=0$ on solving we get x=3/2 or 2/3 now replace x by a/b

so again  putting value of a and b we get final ans is +1 and -1

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