Approximate Expansion of Function

Prove that approximately, when $x$ is very small, $\frac{3{\left(x+\frac{4}{9}\right)}^{\frac{1}{2}}{\left(1-\frac{3}{4}{x}^2\right)}^{\frac{1}{3}}}{2{\left(1+\frac{9}{16}x\right)}^2}=1-\frac{307}{256}{x}^2$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XIV. Binomial Theorem: Any Index, Examples XIV.c, No. 9