<s>[INST] no. and nature of roots of $x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$ The given equation is 
$$x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$$
I took $\log_{2}{x}$ = $t$
and then rewrote the given equation as
$$x^{3t^2 + 4t - 5} = \sqrt{2}$$ 
But I don't know what to do after this.
How will I find the nature and no. of roots?
 [/INST] $$x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$$
$$\log_2 {x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}}} = \log_2 {\sqrt{2}}$$
$$\log_{2}{x} ({\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}})=\frac12$$
$t=\log_2{x}$
$$3t^3+4t^2-5t-2=0$$
$t_1=1.$
Can you finish?
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