Question

The equation $x^{\frac{3}{4}({\log }_{2}x{)}^2+{\log }_{2}x-\frac{5}{4}}=\sqrt{2}$ has

• A. at least one negative solution
• B. exactly one irrational solution
• C. exactly three real solution
• D. two nonreal complex roots

Answer 1

Answer: (B), (C)

The correct options are
B exactly one irrational solution
C exactly three real solution

Given equation$x^{\left[3/4\right(lo{g}_{2}x{)}^2+{\log }_{2}x-5/4]}=\sqrt{2}$
Apply log on both sides and take ${\log }_{2}x=a$
We get $a=1,a=2,a=1/3$
So total three real roots$2^2,2^1,2^{1/3}$
and one irrational $2^{1/3}$