Math Olympiad Problem

`\sqrt{x\sqrt{2x\sqrt{3x}}} =12`

let's do square on Both side we will get

`\left( \sqrt{x\sqrt{2x\sqrt{3x}}}\right)^{2} =12^{2}`

We will get

`\left( x\sqrt{2x\sqrt{3x}}\right) =144`

Let's do square on both sides again

`\left( x\sqrt{2x\sqrt{3x}}\right)^2 =144^2`

We will get

`\left( x^{2}.2x\sqrt{3x}\right)^2 =(12^{4})`

Square it again and we will get

`x^{4}.4x^{2}.3x=(12^{4})^2`

`12x^{7}=12^{8}`

Divide both side with 12 we will

`x^{7} = 12^{7}`

Since power is same we can equate base and hence our answer is

`x=12`

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