Exponential Math Olympiad Problem . Find the value of x

 Math Olympiad Problem `x^{x}=3^{27+2x}`  . Find the Value of  x

Solution

`x^{x}=3^{27+2x}`

We can rewrite the above equation as

`x^{x}=3^{27}.3^{2x}`

Now divide both side by `3^{2x}` we 

`\frac{x^{x}}{3^{2x}}=\frac{3^{27}.3^{2x}}{3^{2x}}`

`\left( \frac{x}{3^{2}} \right)^{x}=3^{27}`

`\left( \frac{x}{9} \right)^{x}=3^{27}`

Now multiply `\frac{1}{9}` power on both sides

`\left( \frac{x}{9} \right)^{x \times \frac{1}{9}}=3^{27 \times \frac{1}{9}} `

`\left( \frac{x}{9} \right)^{\frac{x}{9}}=3^{3} `

if `a^{a} =b^{b} ` then a =b

`\frac{x}{9}=3`

x=27

Answer